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The  Elements  of  the 

Theory  of  Algebraic  Numbers 


BY 


LEGH  WILBER  REID 

Professor  of  Mathematics  in  Haverford  College 


WITH  AN 


INTRODUCTION 


BY 


DAVID   HILBERT 

Professor  of  Mathematics  in  the  University  of  Gottingen 


OF  "HE 

UNIVERSITY 

OF 


Neto  ¥orft 
THE  MACMILLAN    COMPANY 

1910 


«£*£%, 


Copyright  1910 
By  Legh  Wilber  Reid 


PRESS  OF 

the  new  Era  printing  compan* 
lancaster.  pa 


TO 
MY  WIFE. 


PREFACE. 

It  has  been  my  endeavor  in  this  book  to  lead  by  easy  stages  a 
reader,  entirely  unacquainted  with  the  subject,  to  an  appreciation 
of  some  of  the  fundamental  conceptions  in  the  general  theory  of 
algebraic  numbers.  With  this  object  in  view,  I  have  treated  the 
theory  of  rational  integers  more  in  the  manner  of  the  general 
theory  than  is  usual,  and  have  emphasized  those  properties  of 
these  integers  which  find  their  analogues  in  the  general  theory. 
The  same  may  be  said  of  the  general  quadratic  realm,  which  has 
been  treated  rather  as  an  example  of  the  general  realm  of  the 
nth  degree  than  simply  as  of  the  second  degree,  as  little  use  as 
was  possible,  without  too  great  sacrifice  of  simplicity,  being  made 
of  the  special  properties  of  the  quadratic  realm  in  the  proofs. 
The  theorems  and  their  proofs  have  therefore  been  so  formulated 
as  to  be  readily  extendable,  in  most  cases,  to  the  general  realm 
of  the  11th.  degree,  and  it  is  hoped  that  a  student,  who  wishes  to 
continue  the  study  of  the  subject,  will  find  the  reading  of  works 
on  the  general  theory,  such  as  Hilbert's  Bericht  iiber  die  Theorie 
der  Algebraischen  Zahlkorper,  rendered  easier  thereby.  The 
realm  k  ( V  —  I )  has  been  discussed  at  some  length  with  two 
objects  in  view ;  first,  to  show  how  exactly  the  theorems  relating 
to  rational  integers  can  be  carried  over  to  the  integers  of  a  higher 
realm  when  once  the  unique  factorization  theorem  has  been  estab- 
lished; and  second,  to  illustrate,  by  a  brief  account  ofr-Gauss'  work 
in  biquadratic  residues,  the  advantage  gained  by  widening  our  field 
of  operation.  The  proofs  of  the  theorems  relating  to  biquadratic 
residues  have  necessarily  been  omitted  but  the  examples  given  will 
make  the  reader  acquainted  with  their  content.  The  realms 
&(V  —  3)  and  &(V2)  have  been  briefly  discussed  in  order  to 
introduce  the  reader  to  modifications  which  must  be  made  in  our 
conceptions  of  integers  and  units.  In  &(y  —  5),  the  failure  of 
the  unique  factorization  law  is  shown  and  its  restoration  in  terms 
of  ideal  factors  is  foreshadowed. 


VI  PREFACE. 

References  have  been  given  more  with  a  view  to  aiding  the 
student  in  continuing  his  study  of  the  subject  than  to  pointing  out 
the  original  source  of  a  theorem  or  concept. 

The  author  has  adopted  the  term  "  realm  "  as  the  equivalent  of 
korper,  corpus,  campus,  body,  domain  and  field,  as  it  has  the 
advantage,  he  believes,  of  not  having  been  used  in  any  other 
branch  of  mathematics.  It  is  suggested  by  Gauss'  use  of  the 
term  "  Biirgerrecht "  in  connection  with  his  introduction  of  the 
integers  of  k(y/  —  i)  as  his  field  of  operation  (see  p.  218). 

Many  numerical  examples  have  been  given,  especially  in  cases 
involving  ideals,  and  it  is  hoped  that  through  them  the  student 
may  attain  some  familiarity  with  the  methods  of  reckoning  with 
algebraic  numbers.  The  fact  that  the  earlier  discoveries  in  the 
theory  of  numbers  were  made  inductively  inspires  the  belief  that 
such  discoveries  may  also  be  made  in  the  higher  theory,  if  a 
sufficient  amount  of  numerical  material  be  at  hand. 

The  following  is  a  list  of  the  principal  authorities  that  have  been 
consulted,  the  abbreviations  used  in  citation  being  given.  The 
lectures  of  Professor  Hilbert,  mentioned  above,  the  use  of  which 
he  kindly  allowed  me.  Bachmann:  Die  Lehre  von  der  Kreis- 
theilung;  Elemente  der  Zahlentheorie ;  Niedere  Zahlentheorie ; 
Allegmeine  Arithmetik  der  Zahlenkorper.  Borel  et  Drach:  Le- 
cons  sur  la  Theorie  des  Nombres  et  Algebra.  Cahen:  Elements 
de  la  Theorie  des  Nombres,  cited  as  Cahen.  Cayley:  Encyclo- 
paedia Britannica,  9th  ed.,  Vol.  XVII,  pp.  614-624.  Chrystal: 
Algebra.  Dirichlet-Dedekind :  Vorlesungen  iiber  Zahlentheorie, 
4th  ed,  cited  as  Dirichlet-Dedekind.  Gauss :  Disquisitiones  Arith- 
meticae,  Works,  Vol.  I;  Theoria  Residuorum  Biquadraticorum, 
Commentatio  Prima,  Commentatio  Secunda,  Works,  Vol.  II. 
Hilbert:  Bericht  iiber  die  Theorie  der  Algebraischen  Zahlkorper, 
Jahresbericht  der  Deutschen  Mathematiker-Vereinigung,  Vol.  IV, 
cited  as  Hilbert:  Bericht.  Kronecker:  Vorlesungen  iiber  Zahlen- 
theorie. Laurent:  Theorie  des  Nombres,  Ordinaires  et  Alge- 
briques.  Mathews :  Theory  of  Numbers,  cited  as  Mathews ;  also 
Encyclopaedia  Britannica,  Supplement,  Vol.  XXXI.  Minkowski : 
Geometrie  der  Zahlen ;  Diophantische  Approximationen.     H.  J. 


PREFACE.  Vll 

S.  Smith :  Report  on  the  Theory  of  Numbers,  Collected  Mathe- 
matical Papers,  Vol.  I,  pp.  38-364,  cited  as  H.  J.  S.  Smith.  Tsche- 
byscheff :  Theorie  der  Congruenzen.  Weber :  Algebra.  Wertheim ; 
Elemente  der  Zahlentheorie ;  Anfangsgriinde  der  Zahlenlehre. 

In  conclusion,  I  wish  to  express  my  most  sincere  thanks  to 
Professor  Hilbert  for  having  given  me  my  first  interest  in  the 
subject  of  the  theory  of  numbers  by  his  lectures,  which  I  attended 
in  the  winter  semester,  1897-98,  at  Gottingen,  for  his  continued 
interest  in  my  work,  and  for  his  great  kindness  in  writing  an 
introduction  to  this  book.  I  desire  also  to  acknowledge  my 
indebtedness  to  Professor  James  Harkness  of  McGill  Uni- 
versity for  many  helpful  suggestions,  and  to  the  late  Professor 
J.  Edmund  Wright  of  Bryn  Mawr  College  and  my  colleague 
Professor  W.  H.  Jackson  for  valuable  assistance  with  the  proof 
sheets. 

Legh  W.  Reid. 

Haverford  College. 


CONTENTS. 

INTRODUCTION. 
CHAPTER   I. 

Preliminary  Definitions  and  Theorems. 

§  I.  Algebraic  numbers.     Algebraic  integers.     Degree  of  an  algebraic 

number    I 

§  2.  Algebraic  number  realms 3 

§3.  Generation  of  a  realm    3 

§  4.  Degree  of  a  realm.     Conjugate  realm.     Conjugate  numbers 5 

§  5.  Forecast  of  remaining  chapters   5 

CHAPTER   II. 

The  Rational  Realm. 

Divisibility  of  Integers. 

§  1.  Numbers  of  the  rational  realm   7 

§  2.  Integers  of  the  rational  realm 7 

§  3.  Definition  of  divisibility 8 

§  4.  Units  of  the  rational  realm   8 

§  5.  Rational  prime  numbers 9 

§  6.  The  rational  primes  are  infinite  in  number    10 

§  7.  Unique    factorization    theorem    12  - 

§  8.  Divisors  of  an  integer  23 

§  9.  Determination  of  the  highest  power  of  a  prime,  p,  by  which  m ! 

is  divisible 26  * 

tn ' 
§  10.  The    quotient  ,    — —  1   where   m,  =  a  +  &+•••  +  k,  is  an 

a  !  0  !  •  •  •  k ! 

integer   28 

CHAPTER    III. 

The  Rational  Realm. 

Congruences. 

§  1.  Definition.     Elementary  theorems    31 

§  2.  The  function  0  ( m)   2>7 

§  3.  The  product  theorem  for  the  <P- function 45 

§  4.  The   summation  theorem   for  the  0-f unction 46  y 

ix 


X  CONTENTS. 

§  5.  Discussion  of  certain  functional  equations  and  another  derivation 

of  the  general  expression  of  <t>{m)   in  terms  of  in 49 

§  6.  0-f unctions  of  higher  order 54 

§7.  Residue  systems  formed  by  multiplying  the  numbers  of  a  given 

system  by  an  integer  prime  to  the  modulus  56 

§  8.  Fermat's  Theorem  as  generalized  by  Euler 57 

§9.  Congruences    of   condition.     Preliminary   discussion 59 

§  10.  Equivalent  congruences   62 

§11.  Systems  of  congruences.     Equivalent  systems    64 

§  12.  Congruences  in  one  unknown.     Comparison  with  equations   ....  66 

§  13.  Congruences  of  the  first  degree  in  one  unknown 68 

§  14.  Determination  of  an  integer  that  has  certain  residues  with  respect 

to  a  given  series  of  moduli  70 

§  15.  Divisibility  of  one  polynomial  by  another  with  respect  to  a  prime 

modulus.     Common  divisors.     Common  multiples  76 

§  16.  Unit  and  associated  polynomials  with  respect  to  a  prime  modulus. 

Primary  polynomials    JJ 

§  17.  Prime  polynomials  with  respect  to  a  prime  modulus.  Deter- 
mination of  the  prime  polynomials,  mod  p,  of  any  given  degree.  78 
§  18.  Division  of  one  polynomial  by  another  with  respect  to  a  prime 

modulus   ,. 79 

_i§  19.  Congruence  of  two  polynomials  with  respect  to  a  double  modulus.  81 
§  20.  Unique  fractionization  theorem  for  polynomials  with  respect  to 

a  prime  modulus   82 

§21.  Resolution  of  a  polynomial  into  its  prime  factors  with   respect 

to  a  prime  modulus   87 

§  22.     The  general  congruence  of  the  nth  degree  in  one  unknown  and 

with    prime    modulus    88 

§  23.  The  congruence  x4*m)  —  1^0,  mod  m 90 

§  24.  Wilson's   Theorem    , 91 

§  25.  Common  roots  of  two  congruences   92 

§26.  Determination  of  the  multiple  roots  of  a  congruence  with  prime 

modulus     93 

§  27.  Congruences  in  one  unknown  and  with  composite  modulus ....  95 

§  28.  Residues  of  powers 98 

§  29.  Primitive    roots 104 

§  30.  Indices    105 

§  31.  Solution  of  congruences  by  means  of  indices 108 

§  32.  Binomial   congruences no 

§33.  Determination  of  a  primitive  root  of  a  given  prime  number 112 

§  34.  The  congruence  xn  ===  b,  mod  p.     Euler's  criterion 114 


CONTENTS.  XI 

CHAPTER   IV. 

The  Rational  Realm. 

Quadratic  Residues. 

§  i.  The  general  congruence  of  the  second  degree  with  one  unknown.  119 

§  2.  Quadratic  residues  and  non-residues 121 

§  3.  Determination   of  the  quadratic  residues   and  non-residues   of   a 

given   odd  prime  modulus 124 

§  4.  Legendre's   Symbol    127 

§  5.  Determination  of  the  odd  prime  moduli  of  which  a  given  integer 

is  a  quadratic  residue 128 

§  6.  Prime  moduli  of  which  —  1  is  a  quadratic  residue 128 

§7.  Determination   of   a    root   of   the   congruence   x2^ — 1,    mod   p, 

(p  =  4m  -f- 1)   by  means  of  Wilson's  Theorem 129 

§  8.  Gauss'   Lemma    130 

§  9.  Prime  moduli  of  which  2  is  a  quadratic  residue 133 

§  10.  Law   of    reciprocity   for   quadratic    residues 135 

§11.  Determination  of  the  value  of  (a/p)  by  means  of  the  quadratic 

reciprocity  law,  a  being  any  given  integer  and  p  a  prime 144 

§  12.  Determination  of  the  odd  prime  moduli  of  which  a  given  positive 

odd  prime  is  a  quadratic  residue 145 

§  13.  Determination    of   the    odd   prime    moduli   of    which   any   given 

•  integer  is  a  quadratic  residue   147 

§  14.  Other  applications  of  the  quadratic  reciprocity  law 149 

CHAPTER   V. 
The  Realm  k(i). 

§  1.  Numbers  of  k(i).     Conjugate  and  norm  of  a  number 155 

§  2.  Integers  of  k  (»") 157 

§  3.  Basis   of  k  (i) 159 

§  4.  Discriminant  of  k  (i) 161 

§  5.  Divisibility  of  integers  of  k  (i) 162 

§  6.  Units  of  k  (i) .    Associated  integers 163 

§  7.  Prime  numbers  of  k  (*) 165 

§  8.  Unique  factorization  theorem  for  k(i) 167 

§  9.  Classification  of  the  prime  numbers  of  k(i) 177 

§10.  Factorization   of    a   rational   prime   in    k(i)    determined   by   the 

value  of   (d/p) 179 

§  11.  Congruences  in  k(i) 180 

§  12.  The  0-f unction  in  k  (t) 185 

§  13.  Residue  systems  formed  by  multiplying  the  numbers  of  a  given 

system  by  an  integer  prime  to  the  modulus 188 

§14.  The  analogue  for  k(i)   of  Fermat's  Theorem 189 

§  15.  Congruences  of  condition 190 


Xll  CONTENTS. 

§  16.  Two  problems  191 

§  17.  Primary  integers  of  k(i) 193 

§18.  Quadratic  residues  and  the  quadratic  reciprocity  law  in  k(i)..  196 

§  19.  Biquadratic    residues    205 

CHAPTER   VI. 

The  Realm  fc(V-— 3)- 

§  1.  Numbers  of  fc(V  —  3) 218 

§  2.  Integers  of  k (V  — 3) 219 

§3-  Basis  of  k(\/  —  3) ^^ 220 

§  4.  Conjugate  and  norm  of  an  integer  of  &  (V  —  3) 221 

§  5.  Discriminant    of    £  (V  —  3) 221 

§  6.  Divisibility  of  integers  of  &(V —  3) 221 

§7.  Units    of   &(V  —  3).    Associated    integers 222 

§8.  Prime   numbers   of   &(V — 3) 223 

§  9.  Unique  factorization  theorem  for  k (V  —3) 226 

§  10.  Classification  of  the  prime  numbers  of  £(V  — 3) 227 

§11.  Factorization  of  a  rational  prime  in   fc(V  —  3)    determined  by 

the  value  of    (d/p) 229 

§12.  Cubic  residues 230 

CHAPTER  VII. 
The  Realm  k(\f2). 

§  1.  Numbers  of  &(V2) 231 

§2.  Integers    of    &(V2) 231 

§  3.  Discriminant  of   k{  V2) 232 

§  4.  Divisibility  of  integers  of  £(V"2) 232 

§  5.  Units  of  &(V2).    Associated  integers 232 

§  6.  Prime  numbers  of  fc(V2) 235 

§7.  Unique   factorization   theorem   for   &(V2) 236 

§8.  Classification   of  the  prime  numbers   of   k(\j2) 238 

§9.  Factorization  of  a  rational  prime  in  &(V2)    determined  by  the 

value    of    (d/p) 240 

§  10.  Congruences  in  &(V2) 240 

§11.  The  Diophantine  equations  x2 —  2y2  =  ±  1,  x2  —  2y2=±/>,  and 

x2  —  2y2  =  ±  m    240 

CHAPTER   VIII. 

The  Realm  £(\A—  5)- 

§  1.  Numbers  of  fc(V  —  5) 245 

§ 2.  Integers  of  fc(V  —  5) 245 

§3.  Discriminant  of  fc(V  —  5) 245 


CONTENTS.  Xlll 

§  4.  Divisibility  of  integers  of  k( V  —  5) 245 

§  5.  Units    of   k  ( \A— 5)  •    Associated    integers 246 

§6.  Prime   numbers   of   fc(V  —  5) 246 

§7.  Failure  of  the  unique  factorization  theorem  in  &(V  —  5).     Intro- 
duction of  the  ideal 247 

§  8.  Definition  of  an  ideal  of  k(y/  —  5) 257 

§  9.  Equality  of  ideals 258 

§  10.  Principal  and  non-principal  ideals 260 

§  1 1.  Multiplication    of    ideals 261 

§  12.  Divisibility    of    ideals 263 

§  13.  The  unit   ideal 263 

§  14.  Prime   ideals    263 

§  15.  Restoration  of  the  unique  factorization  law  in  terms  of  ideal 

factors    265 


CHAPTER   IX. 
General  Theorems  Concerning  Algebraic  Numbers. 

§  1.  Polynomials  in  a  single  variable , .  268 

§  2.  Numbers    of    a    realm 271 

CHAPTER   X. 
The   General  Quadratic  Realm. 

§  1.  Number  defining  the  realm 280 

§  2.  Numbers    of    the    realm.     Conjugate    and    norm    of    a    number. 

Primitive    and    imprimitive    numbers 281 

§  3.  Discriminant  of  a  number 284 

§  4.  Basis  of  a  quadratic  realm 284 

§  5.  Discriminant  of  the  realm 287 

§ 6.  Determination  of  a  basis  of  k(yjm) 289 

CHAPTER   XI. 
Ideals  of  a  Quadratc  Realm. 

§  1.  Definition.     Numbers    of    an    ideal 293 

§2.  Basis  of  an  ideal.     Canonical  basis.     Principal  and  non-principal 

ideals    294 

§  3.  Conjugate  of  an  ideal 301 

§  4.  Equality  of  ideals 302 

§  5.  Multiplication  of  ideals 302 

§  6.  Divisibility  of  ideals.    The  unit  ideal.     Prime  ideals 303 

§  7.  Unique   factorization  theorem  for  ideals 305 


XIV  CONTENTS. 

CHAPTER  XII. 
Congruences  whose  Moduli  are  Ideals. 

§  i.  Definition.     Elementary  theorems    323 

§  2.  The  norm  of  an  ideal.     Classification  of  the  numbers  of  an  ideal 

with  respect  to  another  ideal 326 

§  3.  Determination  and  classification  of  the  prime  ideals  of  a  quadratic 

realm    339 

§  4.  Resolution  of  any  given  ideal  into  its  prime  factors 348 

§  5.  Determination  of  the  norm  of  any  given  ideal 351 

§  6.  Determination  of  a  basis  of  any  given  ideal 351 

§7.  Determination  of  a  number  a  of  any  ideal  a  such  that   (a) /a  is 

prime  to  a  given  ideal,  m 356 

§  8.  The  0-f unction  for  ideals 358 

§  9.  Residue  systems  formed  by  multiplying  the  numbers  of  a  given 

system  by  an  integer  prime  to  the  modulus 367 

§10.  The  analogue  for  ideals  of  Fermat's  Theorem 368 

§  1 1.  Congruences    of    condition 369 

§  12.  Equivalent  congruences   372 

§  13.  Congruences  in  one  unknown  with  ideal  moduli 374 

§  14.  The  general  congruence  of  first  degree  with  one  unknown ....  375 
§  15.  Divisibility  of  one  polynomial  by  another  with  respect  to  a  prime 

ideal  modulus.     Common  divisors.     Common  multiples 380 

§  16.  Unit  and  associated  polynomials  with  respect  to  a  prime  ideal 

modulus.     Primary  polynomials    380 

§  17.  Prime  polynomials  with  respect  to  a  prime  ideal  modulus.  De- 
termination  of   the  prime   polynomials,   mod   p,   of   any  given 

degree 381 

§  18.  Division  of  one  polynomial  by  another  with  respect  to  a  prime 

ideal  modulus   382 

§  19.  Unique  f ractorization  theorem   for  polynomials  with  respect  to 

a  prime  ideal  modulus 382 

§20.  The  general  congruence  of  the  nth.  degree  in  one  unknown  and 

with  prime  ideal  modulus 385 

§21.  The  congruence  #*(m> — i==o,  mod  m 387 

§22.  The  analogue  for  ideals  of  Wilson's  Theorem 388 

§  23.  Common  roots  of  two  congruences 389 

§  24.  Determination  of  the  multiple  roots  of  a  congruence  with  prime 

ideal  modulus 390 

§  25.  Solution  of  congruences  in   one  unknown  and  with  composite 

modulus   391 

§  26.  Residues  of  powers  for  ideal  moduli 392 

§27.  Primitive  numbers  with  respect  to  a  prime  ideal  modulus 398 

§  28.  Indices    398 

§  29.  Solution  of  congruences  by  means  of  indices 400 


CONTENTS.  XV 

CHAPTER   XIII. 

The  Units  of  the  General  Quadratic  Realm. 

§  i.  Definition    403 

§  2.  Units  of  an  imaginary  quadratic  realm 404 

§  3.  Units  of  a  real  quadratic  realm 405 

§  4.  Determination  of  the  fundamental  unit 420 

§  5.  Pell's  Equation   423 

CHAPTER   XIV. 

The  Ideal  Classes  of  a  Quadratic  Realm. 

§  1.  Equivalence  of  ideals 427 

§  2.  Ideal  classes 432 

§  3.  The  class  number  of  a  quadratic  realm 434 

Index 452 


ERRATA. 

Page  2.J,  line  5,  for  "fxn  read  "p." 
Page  172,  line  5,  for  'V3  = —  4  —  p"  read  "  fis  =  2. 

"  p         o 


Page  344,  line  10,  for 


P     o    " 
o     I 


read 


p-\ 


11     2 

Page    356,    line    29,     for     "(7,     3  +  V~ 5) "     read     "(7, 
3-V-5)." 

Page  392,  line  3  of  fine  print,  for  "of  (x)  "  read  "f(x)." 


INTRODUCTION.    • 

Die  Zahlentheorie  ist  ein  herrlicher  Bau,  erschaffen  und  auf- 
gefiihrt  von  Mannern  die  zu  den  glanzendsten  Forschern  im 
Bereiche  der  mathematischen  Wissenschaften  gehoren:  Fermat, 
Euler,  Lagrange,  Legendre,  Gauss,  Jacobi,  Dirichlet,  Hermite, 
Kummer,  Dedekind  und  Kronecker ;  Alle  diese  Manner  haben  in 
den  begeistersten  Worten  ihrer  hohen  Meinung  uber  die  Zahlen- 
theorie Ausdruck  gegeben  und  bis  heute  giebt  es  wohl  keins 
Wissenschaft,  von  deren  Ruhme  ihre  Jiinger  so  erfiillt  sind,  wie 
von  der  Zahlentheorie.  Man  preist  an  der  Zahlentheorie  die 
Einfachheit  ihrer  Grundlagen,  die  Genauigkeit  ihrer  Begriffe  und 
die  Reinheit  ihrer  Wahrheiten;  man  ruhmt  sie  als  das  Vorbild 
fur  die  anderen  Wissenschaften,  als  die  tiefste  unversiegbare 
Quelle  aller  mathematischen  Erkenntniss  und  als  reiche  Spenderin 
von  Anregungen  fur  andere  mathematische  Forschungsgebietc 
wie  Algebra,  Funktionentheorie,  Analysis  und  Geometric  Dazu 
kommt,  dass  die  Zahlentheorie  vom  Wechsel  der  Mode  unab- 
hangig  ist  und  dort  nicht  wie  oft  in  anderen  Wissensgebieten, 
bald  die  eine  Auffassung  oder  Methode  iibermassig  sich  auf- 
baus§ht,  bald  zu  anderer  Zeit  unverdiente  Zuriicksetzung  erf ahrt ; 
in  der  Zahlentheorie  ist  oft  das  alteste  Problem  noch  heute 
modern,  wie  ein  echtes  Kunstwerk  aus  der  Vergangenheit. 

Und  dennoch  ist  jetz  wie  friiher  wahr,  wo  ruber  Gauss  und 
Dirichlet  klagten,  dass  nur  eine  geringe  Anzahl  von  Mathe- 
matikern  zu  einer  eingehenden  Beschaftigung  mit  der  Zahlen- 
theorie und  zu  einem  vollen  und  freien  Genusse  ihrer  Schonheit 
gelangt.  Zumal  ausserhalb  Deutschlands  und  unter  der  heran 
wachsenden  mathematischen  Jugend  ist  arithmetisches  Wissen 
nur  wenig  verbreitet. 

Jeder  Liebhaber  der  Zahlentheorie  wird  wunschen,  dass  die 
Zahlentheorie  gleichmassig  ein  Besitz  aller  Nationen  sei  und 
gerade  besonders  unter  der  jungen  Generation,  der  die  Zukunft 

xvii 


XV111  INTRODUCTION. 

gehort,  Pflege  und  Verbreitung  finde.  Das  vorliegende  Buch 
steckt  sich  dieses  Ziel :  Moge  es  dasselbe  erreichen,  indem  es  nicht 
nur  dazu  beitrage,  dass  die  Elemente  der  Zahlentheorie  Gemein- 
gut  aller  Mathematiker  werden,  sondern,  indem  es  auch  zugleich 
als  Einfuhrung  und  Erleichterung  zum  Studium  der  darin  ge- 
nannten  Originalwerke  diene,  sowie  zur  selbstandigen  Betha- 
tigung  der  Zahlentheorie  anrege.  Bei  der  liebevollen  Vertiefung 
des  Verfassers  in  die  Zahlentheorie  und  bei  dem  hingebenden 
Verstandniss,  mit  dem  der  Verfasser  in  das  Wesen  derselben 
eingedrungen  ist,  durfen  wir  auf  die  Erfullung  dieses  Wunsches 
bauen. 

David  Hilbert. 
Gottingen,  io,  Marz,  1907. 


TRANSLATION. 

The  theory  of  numbers  is  a  magnificent  structure,  created  and  developed 
by  men  who  belong  among  the  most  brilliant  investigators  in  the  domain 
of  the  mathematical  sciences :  Fermat,  Euler,  Lagrange,  Legendre,  Gauss, 
Jacobi,  Dirichlet,  Hermite,  Kummer,  Dedekind  and  Kronecker.  All  these 
men  have  expressed  their  high  opinion  respecting  the  theory  of  numbers  in 
the  most  enthusiastic  words  and  up  to  the  present  there  is  indeed  no 
science  so  highly  praised  by  its  devotees  as  is  the  theory  of  numbers.  In 
the  theory  of  numbers,  we  value  the  simplicity  of  its  foundations,  the 
exactness  of  its  conceptions  and  the  purity  of  its  truths ;  we  extol  it  as 
the  pattern  for  the  other  sciences,  as  the  deepest,  the  inexhaustible  source 
of  all  mathematical  knowledge,  prodigal  of  incitements  to  investigation  in 
other  departments  of  mathematics,  such  as  algebra,  the  theory  of  func- 
tions, analysis  and  geometry. 

Moreover,  the  theory  of  numbers  is  independent  of  the  change  of 
fashion  and  in  it  we  do  not  see,  as  is  often  the  case  in  other  depart- 
ments of  knowledge,  a  conception  or  method  at  one  time  given  undue 
prominence,  at  another  suffering  undeserved  neglect;  in  the  theory  of 
numbers  the  oldest  problem  is  often  to-day  modern,  like  a  genuine 
work  of  art  from  the  past.  Nevertheless  it  is  true  now  as  formerly,  a 
fact  which  Gauss  and  Dirichlet  lamented,  that  only  a  small  number  of 
mathematicians  busy  themselves  deeply  with  the  theory  of  numbers  and 
attain  to  a  full  enjoyment  of  its  beauty.  Especially  outside  of  Germany 
and  among  the  younger  mathematicians  arithmetical  knowledge  is  little 
disseminated.  Every  devotee  of  the  theory  of  numbers  will  desire  that  it 
shall  be  equally  a  possession  of  all  nations  and  be  cultivated  and  spread 
abroad,   especially   among  the  younger   generation   to   whom   the   future 


INTRODUCTION.  XIX 

belongs.  Such  is  the  aim  of  this  book.  May  it  reach  this  goal,  not  only 
by  helping  to  make  the  elements  of  the  theory  of  numbers  the  common 
property  of  all  mathematicians,  but  also  by  serving  as  an  introduction  to 
the  original  works  to  which  reference  is  made,  and  by  inciting  to  inde- 
pendent activity  in  the  field  of  the  theory  of  numbers.  On  account  of 
the  devoted  absorption  of  the  author  in  the  theory  of  numbers  and  the 
comprehensive  understanding  with  which  he  has  penetrated  into  its  nature, 
we  may  rely  upon  the  fulfilment  of  this  wish. 


CHAPTER   I. 

Preliminary  Definitions  and  Theorems. 

§  i.  Algebraic  Numbers.  Algebraic  Integers.  Degree  of  an 
Algebraic  Number. 

It  will  be  assumed  in  this  book  that  the  complex  number  system 
has  been  built  up  and  that  the  laws  to  which  the  four  fundamental 
operations  of  algebra  are  subject  have  been  demonstrated  to  hold 
when  these  operations  are  performed  upon  any  numbers  of  this 
system. 

We  shall  occupy  ourselves  with  certain  properties  of  a  special 
class  of  these  numbers,  known  as  algebraic  numbers,  these  prop- 
erties flowing  in  the  greater  part  from  the  relation  in  which  two 
numbers  stand  to  one  another  when  one  is  said  to  be  a  divisor  of 
the  other.     We  proceed  to  define  an  algebraic  number. 

A  number,  a,  is  said  to  be  an  algebraic  number  when  it  satisfies 
an  equation  of  the  form 

xn  -f  a1  xn~x  +  •  •  •  +  On_t  x  +  an  =  o  i) 

where  alt  a2,  •  •  •,  On  are  rational  numbers.  We  shall  call  an  equa- 
tion of  form  i)  a  rational  equation.  The  simplest  algebraic 
numbers  are  evidently  the  rational  numbers.  An  algebraic  num- 
ber is  said  to  be  an  algebraic  integer  or  briefly  an  integer,  when 
it  satisfies  an  equation  of  the  form  i)  whose  coefficients,  alt  a2, 
•  •  •,  an,  are  rational  integers.  The  simplest  algebraic  integers  are 
the  positive  and  negative  natural  numbers.  An  algebraic  number, 
a,  evidently  satisfies  an  infinite  number  of  rational  equations,  for 
if  a  satisfy  i),  it  also  satisfies  any  equation  formed  by  multiplying 
i)  by  an  integral  function  of  x  of  the  form 

#•  +  bx  xm~x  +  •  •  •  +  bm_x  x  +  bm, 

where  blt-  ■  •,  bm  are  rational  numbers,  and  this  equation  will  be  of 
the  form  i).     There  will  be  however  among  all  these  rational 


2  PRELIMINARY   DEFINITIONS   AND   THEOREMS. 

equations  satisfied  by  a,  one  and  only  one  of  lowest  degree,  /. 
For  suppose  that  a  satisfied  two  different  rational  equations  of  the 
/th  degree,  /  being  the  degree  of  the  rational  equation  of  lowest 
degree  satisfied  by  a,  and  let  these  equations  be 

xl  -f  a^x1-1  +  •  •  •  +  ai  =  o  2) 

xi  +  biXi-i  +  --.  +  bi  =  o  3) 

Then  a  will  satisfy  the  equation  formed  by  subtracting  3)  from 

2)  ;  that  is,         (ax —  b^x1-1  -f-  ••  •  +  a\ —  bi  =  o  >  4) 

Unless  4)  be  identically  zero,  a  satisfies  a  rational  equation  of 
degree  lower  than  the  /th,  which  is  contrary  to  our  original  sup- 
position. Therefore  4)  is  identically  o,  and  2)  and  3)  are  the 
same  equation.  Hence  a  satisfies  only  one  rational  equation  of 
the  /th  degree. 

This  equation  is  irreducible ;  that  is,  its  first  member  can  not 
be  resolved  into  factors  of  lower  degree  in  x,  with  rational  coeffi- 
cients ;  for  if 
x  1  +  aiX  1-1  +  . . .  +  a,  =  (xn  +  blXn-i  +  . . .  +  bh) 

X  (^  +  <vtr*-1  +  ---  +  ck), 

where  blt  •••,  bh,  cx,  •••,  cfc  are  rational  numbers,  a  would  satisfy 
one  of  the  rational  equations 

xh  +  b^-1  -f- 1-  bh  =  o ;  xk  +  c^v*-1  +  •  •  •  +  ck  =  o. 

This  is,  however,  impossible  since  both  of  these  equations  are  of 
lower  degree  than  the  /th.  Hence  the  rational  equation  of  lowest 
degree,  which  a  satisfies,  is  irreducible.  If  /  be  the  degree  of 
this  equation,  a  is  said  to  be  an  algebraic  number  of  the  /th 
degree. 

Theorem  i.     If  a  be  an  algebraic  number , 

fx(x)  =xl  +  axxl-x  +  •••  +  fli==d 

the  single  rational  equation  of  lowest  degree  which  a  satisfies, 

and  f2  (x)  =  xm  +  b^v™-1  +  •  •  •  +  bm  =  o 

any  other  rational,  equation  satisfied  by  a,  then  fx(x)  is  a  divisor 
off2(x). 


PRELIMINARY   DEFINITIONS   AND   THEOREMS.  3 

We  can  always  put  f2(x)  in  the  form 

where  f^ix)  and  f4(x)  are  rational  integral  functions  of  x  whose 
coefficients  are  rational  integers  and  f4(x)  of  lower  degree  than 
f1(x).     Substituting  a  for  x  in  2)  we  have 

/2(a)=/3(a)-/1(a)+/4(a), 

whence,  since  f2(a)=o,  and  /i(a)=o,  f*(.CL)=o\  that  is,  unless 
/4(^r)  is  identically  o,  a  will  satisfy  a  rational  equation,  /4(^)  =0 
of  lower  degree  than  the  /th.  But  this  is  contrary  to  our  original 
hypothesis.  Hence  f4(x)  is  identically  zero,  and  fx(x)  is  there- 
fore a  divisor  of  f2(x).  />,!  J- 

We  shall  see  later  (Chap.  II,  Th.  4)  that  the  rational  equation 
of  lowest  degree  which  an  algebraic  number,  a,  satisfies,  deter- 
mines the  question  whether  or  not  a  is  an  algebraic  integer;  that 
is,  that  the  coefficients  of  the  single  rational  equation  of  lowest 
degree,  which  an  algebraic  number,  a,  satisfies,  shall  be  integers, 
is  a  necessary  as  well  as  sufficient  condition  for  a  to  be  an  alge- 
braic integer. 

§  2.    Algebraic  Number  Realms. 

A  system  of  algebraic  numbers  is  called  a  number  realm  or 
briefly  a  realm,  if  the  sum,  difference,  product  and  quotient  of 
every  two  numbers  of  the  system,  excluding  division  by  o,  are 
numbers  of  the  system;  that  is,  if  the  system  is  invariant  with 
respect  to  these  four  operations. 

The  simplest  example  of  a  realm  is  the  system  of  all  rational 
numbers,  for  evidently  the  sum,  difference,  product  and  quotient 
of  any  two  rational  numbers  are  rational  numbers.  Another  ex- 
ample is  the  system  of  numbers  of  the  form  x-\-  y  y  —  T>  where 
x  and  y  take  all  rational  values.  For  the  sum,  difference,  product 
and  quotient  of  any  two  of  these  numbers  are  numbers  of  this 
form. 

§  3.     Generation  of  a  Realm. 

If  a  be  any  algebraic  number,  the  system  consisting  of  all  num- 
bers, which  can  be  formed  by  repeated  performance  upon  a  of  the 


4  PRELIMINARY   DEFINITIONS    AND   THEOREMS. 

four  fundamental  reckoning  operations,  that  is,  the  system  con- 
sisting of  all  rational  functions  of  a  with  rational  coefficients,  will 
be  a  realm. 

For  the  sum,  difference,  product  and  quotient  of  any  two  ra- 
tional functions  of  a  are  rational  functions  of  a  and  hence  num- 
bers of  the  system. 

We  say  that  a  generates  this  realm.  We  say  also  that  a  defines 
the  realm  and  denote  the  latter  by  k(a).  The  rational  realm  can 
be  generated  by  any  rational  number,  a ;  for  a  divided  by  a  gives 
I,  and  from  I  by  repeated  additions  and  subtractions  of  I,  we  can 
obtain  all  rational  integers,  and  from  them  by  division  all  rational 
fractions.  As  the  number  defining  the  rational  realm  we  generally 
take  i,  thus  denoting  the  realm  by  k(i).  More  usually,  how- 
ever, the  rational  realm  is  denoted  by  the  letter  R.  The  realm 
given  as  the  second  example  in  the  last  paragraph  can  be  generated 
by  V — J  ;  f°r  V — !  divided  by  V — T  gives  i,  and  from  I  we 
can  generate  the  rational  realm  and  then  by  multiplying  V —  I 
by  all  rational  numbers  in  turn  and  adding  to  each  of  these 
products  each  rational  number  in  turn,  we  obtain  all  numbers  of 
the  form  x  -f-  yy/ — i,  where  x  and  y  take  all  rational  values. 
This  realm  is  therefore  denoted  by  k(  V —  i).  We  have  seen  in 
the  last  example  that  among  the  numbers  of  k  (  V —  i )  are  found 
all  the  numbers  of  the  rational  realm.  It  may  be  easily  seen  that 
this  is  true  of  every  realm ;  that  is,  every  realm  contains  R ;  for  if 
o>  be  any  number,  w  divided  by  w  gives  i,  and  from  i  we  can 
generate  R.  It  is  well  to  observe  that,  although  V —  i  is  the 
number  which  most  conveniently  defines  k  ( V —  I )  and  is  indeed 
the  one  usually  selected,  it  is  not  the  only  number  that  will  serve 
this  purpose.  We  see,  on  the  contrary,  that  this  realm  can  be 
generated  by  any  number  of  the  form  a  -f-  b  V —  I  where  a 
and  b  are  rational  numbers,  and  b  =f=  o ;  that  is,  k  ( V —  1 )  and 
k{a-\-by/ — 1)  are  identical;  for  since  k(a-\-by/ — 1)  con- 
tains R,  it  contains  a  and  b  and  hence  -^ — ^-r ,  =  y/ — 1. 

Therefore  k(a-\-by/ — 1)  contains  all  numbers  of  k(y/— Y). 
Moreover  since  &(V — 1)   contains  a-\-b\/ — 1,  it  contains  all 


PRELIMINARY   DEFINITIONS   AND   THEOREMS.  5 

numbers  of  k{a-\-b\/  —  i).  Hence  k ( V  —  I )  is  identical  with 
k(a-\-by — 1).  It  may  be  shown  similarly  that  any  realm 
may  be  defined  by  any  one  of  an  infinite  number  of  its  num- 
bers; as,  for  example,  if  a  be  any  algebraic  number,  k(a)  and 
k(a-\-ba),  where  a  and  b  are  rational  numbers,  and  &=f=o  are 
identical.  A  realm  may  be  generated  by  any  number  of  algebraic 
numbers.  If  a,  /?,  •  •  •,  X  are  a  finite  number  of  algebraic  numbers, 
the  system  consisting  of  all  rational  functions  of  these  numbers 
with  rational  coefficients  is  a  realm  which  we  denote  by  k(a,  fi, 
■••, A).  It  can  be  shown,  however,  that  in  every  realm  k(a,  /?, 
••••, A)  we  can  find  a  number  6  such  that  k(a,  /?,  •••,  \)=k{6). 
We  shall  not  prove  this,  as  all  realms  discussed  in  this  book  will 
be  defined  by  a  single  number. 

§  4.  Degree  of  a  Realm.  Conjugate  Realms.  Conjugate 
Numbers. 

If  the  rational  equation  of  lowest  degree  which  a  satisfies  be 

xn  +  a^xn-x  -f  •  •  •  +  an  =  o  1) 

then  k(a)  is  said  to  be  of  the  nth.  degree.  That  is,  the  degree  of 
a_realm  is  the  degree  of  the  number  defining  the  realm.  Thus 
&(V — 1)  is  of  the  second  degree,  since  the  rational  equation  of 
lowest  degree  which  V —  1  satisfies  is  x2  +  1  =  o.  Likewise 
£(1/2)  is  of  the  third  degree.  There  is  evidently  only  one  realm 
of  the  first  degree  k(i),  but  an  infinite  number  of  all  other  de- 
grees. If  the  remaining  roots  of  1 )  be  a',  a",  •  •  •,  a(n_1),  then  n —  1 
realms  k(a'),  k(a")f  •••,  ^(a(n_1))   are  called  the  conjugates  of 

If  6  be  any  number  of  k(a),  it  is  a  rational  function  of  a,  which 
we  may  denote  by  r{a).  Then  0'  =  r(a')>  0"  =  r(a"),  •••, 
0(n_1)=r(a(n~1)),  which  are  derived  from  0  by  the  substitutions 
a: a',  a: a",  •••,a:a(n_1),  are  called  the  conjugates  of  0. 

§  5.     Forecast  of  Remaining  Chapters. 

We  shall  consider  now  several  special  realms.  In  each  we  shall 
find  an  infinite  number  of  algebraic  integers,  the  study  of  whose 
properties  in  their  mutual  relations  will  be  our  task.     It  will  be 


6  PRELIMINARY   DEFINITIONS    AND   THEOREMS. 

observed  that  the  properties  of  an  integer  depend  upon  the  realm 
in  which  it  is  considered  to  lie.  Thus  the  integer  5  is  unfavor- 
able in  R  and  in  k( V — 3),  but  in  k(y/ — 1)  it  is  the  product  of 
two  integers,  2  -f-  V —  1  and  2  —  V —  1. 

The  realms  will  be  taken  up  in  the  order  of  their  degrees. 
That  is,  the  first  to  be  studied  will  be  R,  which  is,  as  has  been 
already  said,  the  only  realm  of  the  first  degree.  We  shall  then 
take  up  in  turn  four  special  examples  of  quadratic  realms, 
£(V — 1),  &(V — 3),  £(V2)  and  &(V — 5).  In  the  cases  of 
&(V — 1),  fc(V — 3)  and  k(\/2),  we  shall  see  that,  with  the 
introduction  of  a  few  new  conceptions,  the  integers  of  these 
realms  obey  in  their  relations  to  each  other  laws  almost  identical 
with  those  governing  the  integers  of  R. 

In  the  case  of  &(V — 5)  we  shall  observe  an  important  differ- 
ence, and  at  first  sight  it  will  seem  that  our  old  laws  have  no 
analogues  in  this  realm.  By  the  introduction,  however,  of  the 
conception  of  the  ideal  number  not  only  will  the  difficulties  of  this 
particular  realm  be  overcome,  but  we  shall  be  able  to  establish 
in  terms  of  these  jdeal  numbers  general  laws  for  the  mutual  rela- 
tions of  the  integers  of  any  quadratic  realm,  which  are  analogous 
to  those  already  found  for  the  integers  of  the  special  realms  ex- 
amined. Furthermore  the  larger  part  of  the  theorems  proved 
for  the  integers  of  the  general  quadratic  realm  hold  for  the  in- 
tegers of  a  realm  of  any  degree  whatever. 


CHAPTER   II. 
The  Rational  Realm. 

divisibility  of  integers. 

§  i.     The  Numbers  of  the  Rational  Realm. 

The  rational  realm  consists  of  the  system  of  rational  numbers, 
any  one  of  which,  except  o,  may  be  taken  to  define  it.  It  is 
usually  denoted  by  k(i)  or  simply  R.  The  absolute  value  of  a 
number,  m,  of  R  is  m  taken  positively  and  is  denoted  by  |  m  \ .     Thus 

l±5l=5- 
The  absolute  value  of  a  number  is  used  when  the  result  of  an 
enumeration  is  to  be  expressed  as  a  function  of  this  number. 

§  2.    Integers  of  the  Rational  Realm. 

The  positive  and  the  negative  rational  integers  are  evidently 
integers  of  R,  for  they  satisfy  equations  of  the  form  ;r-|-a  =  o, 
where  a  is  a  rational  integer.  The  sum,  difference  and  product 
of  any  two  rational  integers  are  seen  to  be  integers.  The  ques- 
tion will  at  once  be  asked,  are  these  all  the  numbers  of  the  rational 
realm  which  are  algebraic  integers  under  the  definition  given  of 
an  algebraic  integer  (Chap.  I,  §  i).  That  is,  although  a  rational 
fraction,  b/c,  where  b  is  not  divisible  by  c  evidently  cannot  satisfy 
an  equation  of  the  form  x-\-a  =  o,  where  a  is  a  rational  integer, 
we  have  not  yet  shown  that  b/c  cannot  satisfy  an  equation  of 
higher  degree  than  the  first  and  of  the  form 

xn  +  axxn~x  -f  •  •  •  -+-  an  =  o, 

where  ax,a2,  ••-,(!„  are  rational  integers. 

To  show  this,  it  is  necessary  to  prove  first  that  a  rational  integer 
can  be  resolved  in  one  and  only  one  way  into  prime  factors. 
Therefore,  until  we  have  proved  this  theorem,  the  integers  with 
which  we  are  dealing  should  be  looked  upon  as  merely  the  ordi- 
nary rational  integers.     When  we  have  proved  the  above  theorem 

7 


8  THE   RATIONAL   REALM INTEGERS. 

we  shall  see  that  the  system  of  rational  integers  and  the  system 
of  integers  of  R  are  coextensive. 

§  3.     Definition  of  Divisibility. 

An  integer,  a,  is  said  to  be  divisible  by  an  integer,  b,  when  there 
exists  an  integer,  c,  such  that  a=bc;  then  b  and  c  are  said  to  be 
divisors,  or  factors,  of  a  and  a  is  said  to  be  a  multiple  of  b  and  c. 
Furthermore,  a  is  said  to  be  resolved  into  the  factors  b  and  c,  or 
to  be  factored. 

We  have,  as  direct  consequences  of  the  definition  of  divisibility 
and  the  fact  that  the  sum,  difference  and  product  of  any  two 
integers  are  integers,  the  following: 

i.  If  a  be  a  multiple  of  b,  and  b  a  multiple  of  c,  a  is  a  multiple 
of  c.  For  since  a  is  a  multiple  of  b,  we  have  a  =  a1b,  and 
since  b  is  a  multiple  of  c,  b  =  bxc.  From  which  it  follows  that 
a  =  axbxc.  Hence  a  is  a  multiple  of  c.  In  general  if  each  integer 
of  a  series  a,  b,  c,  d,  •  •  •,  be  a  multiple  of  the  one  next  following, 
each  integer  is  a  multiple  of  all  that  follow  it ;  that  is,  if  a  be  a 
multiple  of  b,  b  a  multiple  of  c,  c  a  multiple  of  d,  etc.,  a  is  a  mul- 
tiple of  b,  c,  d,  •  -  •,  b  a  multiple  of  c,  d,  •  •  -,  etc. 

ii.  //  two  integers  a  and  b  be  multiples  of  an  integer  c,  a  -f-  b 
and  a  —  b  are  multiples  of  c.  If  two  or  more  integers  a,  b,  c,  ••• 
be  each  divisible  by  an  integer  m,  m  is  said  to  be  a  common 
divisor  or  common  factor  of  a,  b,  c,  ••  •.  If  an  integer,  m,  be  a 
multiple  of  two  or  more  integers,  a,  b,  c,  •••,  m  is  said  to  be  a 
common  multiple  of  a,  b,  c,-  -,1 

§  4.    Units  of  the  Rational  Realm. 

There  are  two  integers,  1  and  —  1,  which  are  divisors  of  every 
rational  integer  and  they  are  the  only  rational  integers  that  enjoy 
this  property. 

We  call  1  and  —  1  the  units  of  R. 

Any  integer  which  is  divisible  by  m  is  also  divisible  by  — m; 
hence  any  two  integers  which  differ  only  by  a  unit  factor  are 
considered  as  identical  in  all  questions  of  divisibility.     We  say 

throughout  this  book  the  letters  of  the  Latin  alphabet  will  always 
denote  rational  numbers,  unless  there  be  a  direct  statement  to  the  contrary. 


THE   RATIONAL   REALM INTEGERS.  9 

that  two  such  integers  are  associated,  and  call  either  one  the  asso- 
ciate of  the  other.  Two  integers,  a  and  b,  each  of  which  divides 
the  other,  are  associates,  for  if  a  =  cb  and  b  =  da  where  c  and  d 
are  integers,  then  cd=i,  and  hence  c=±i.  Two  integers 
whose  absolute  values  are  the  same  are  evidently  associates.  For 
the  sake  of  generality  we  consider  an  integer  as  associated  with 
itself. 

Thus  the  associates  of  5  are  5  and  —  5  since 

5  =  1-5  and— 5— —1-5. 

The  factorizations  of  30, 

30  =  2-3-5, 
=  —  2-  —  3-5, 

=  —  2-3*  —  5, 
=  2  —  3-  —  5, 

are  looked  upon  as  identical,  since  they  differ  only  by  the  replace- 
ment of  one  or  more  of  the  factors  by  their  associates. 

Two  integers  with  no  common  divisors  other  than  units  are 
said  to  be  prime  to  each  other. 

Under  this  definition  the  units  are  considered  prime  to  every 
integer  including  themselves. 

if  i'«i-i»i 

a  and  b  are  associates,  and  it  follows  therefore  that  if  a  be 

prime  to  b  |  a  |  =(=  |  b  | 

unless  a  and  b  be  units. 

A  system  of  integers  such  that  no  two  of  them  have  common 
divisors  other  than  the  units  are  said  to  be  prime  each  to  each. 

§  5.    Rational  Prime  Numbers. 

Any  integer,  p,  that  is  not  a  unit  and  that  has  no  divisors  other 
than  p  and  — p,  1  and  — 1,  that  is,  than  its  associates  and  the 
units,  is  called  a  prime  number  or,  briefly,  a  prime. 

The  units  are  not  considered  to  be  prime  numbers,  because  many 
of  the  theorems  relating  to  prime  numbers  will  be  found  not  to 
hold  for  them. 


IO  THE    RATIONAL   REALM INTEGERS. 

Every  integer,  m,  with  divisors  other  than  m,  — m,  I,  — I  is 
called  a  composite  number.  We  can  obtain  the  positive  prime 
numbers  less  than  any  given  positive  integer,  m,  as  follows :  The 
only  even  one  is  2.     We  write  down  then  the  odd  integers  smaller 

than  m,  3,  5,  7,  9,  11,  13,  15,  17,  19,  21,  •••, 

and  remove  from  the  series  those  which  are  composite.  To  do 
this  strike  out,  counting  from  3,  the  3d,  6th,  9th,  •  •  •  integers ; 
that  is,  9,  15,  21,  ••*.  Then  counting  from  5,  strike  out  the  5th, 
10th,  •••  integers;  that  is,  15,  25,  •••,  counting  integers  already 
struck  out,  and  in  general,  if  p  be  the  smallest  integer  not  struck 
out,  excluding  those  whose  multiples  have  been  struck  out,  we 
strike  out  the  pth,  2pth,  3pth,  •  •  •  integers,  counting  from  p ;  that 
is,  all  multiples  of  p  except  p.  The  integers  not  struck  out  are 
the  positive  primes  smaller  than  m. 

This  method  is  known  as  the  Sieve  of  Eratosthenes.  It  is, 
however,  not  necessary  to  carry  out  the  process  for  every  prime,  p, 
smaller  than  m ;  for  every  composite  number,  mlf  smaller  than  p2, 
will  have  been  struck  out  as  a  multiple  of  a  prime  smaller  than  p, 
since  if  m1  be  less  than  p2,  it  contains  as  a  factor  a  prime  less  than 
p.  The  greatest  value  of  p  for  which  the  process  must  be  car- 
ried out  is  therefore  the  greatest  prime  not  greater  than  s/m. 

The  positive  primes  less  than  100  are:  2,  3,  5,  7,  11,  13,  17,  19, 
23,  29,  31,  37,  41,  43,  47,  53,  59,  61,  67,  71,  73,  79,  83,  89,  97. 

Ex.  1.  Show  that  every  rational  prime,  except  2,  is  either  of  the  form 
4«  —  1  or  4*1  -\-  1. 

Ex.  2.  Show  that  every  rational  prime,  except  2  and  3,  is  either  of  the 
form  6n  —  1  or  6n  +  1. 

§  6.    The  Rational  Primes  are  Infinite  in  Number. 

The  proof  of  this  theorem  as  given  by  Euclid  (Elements,  Book 
IX,  Prop.  20)  is  the  following :  Let  us  suppose  that  there  are  only  a 
finite  number  of  positive  primes,  p  being  the  greatest.  Multiply 
these  primes  together  and  add  1  to  the  product,  forming  the  number 

It  is  evident  that  N  is  not  divisible  by  any  of  the  primes  2,  3,  5, 
•••,  p.     Hence  N  is  either  a  prime  itself,  or  contains  as  a  factor 


THE   RATIONAL   REALM INTEGERS.  I  I 

a  prime  greater  than  p.  In  either  case  there  exists  a  prime  greater 
than  p,  which  contradicts  our  original  assumption.  Hence  the 
number  of  rational  primes  is  infinite. 

This  proof  of  Euclid's  tells  us  far  more  than  merely  that  the 
rational  primes  are  infinite  in  number,  for  if  2,  3,  •  •  •,  p  be  the  n 
smallest  positive  primes  it  gives  a  limit,  p-\-i  to  2-y-p  -\-  1, 
within  which  a  prime  greater  than  p  must  lie.  To  bring  out 
clearly  what  has  been  proved  we  may  state  the  theorem  as  follows : 
//  2,  2,  '",  P  be  the  n  smallest  positive  primes,  then  there  is  a 
prime  greater  than  p  among  tlve  numbers  p  -\-i,  •••,2-3  -•  p  -\-i 
and  consequently  the  rational  primes  are  infinite  in  number.  For 
example,  2,  3,  5,  7,  being  all  the  positive  rational  primes  not 
greater  than  7,  there  is  certainly  one  prime  greater  than  7  among 
the  numbers  8,9,  •••,3:3"5,7-+  I. 

After  it  became  known  that  the  rational  primes  are  infinite  in 
number,  the  attention  of  investigators  was  turned  to  the  question 
whether,  if  from  the  positive  integers  a  series  be  selected  which 
form  an  arithmetical  progression,  as  for  example  1,  5,  9,  13,  •••, 
or  3,  7,  11,  15,  ■••,  there  are  in  every  such  series  an  infinite  number 
of  primes.  Proofs  showing  that  this  is  true  of  the  two  series 
given  will  be  found  in  this  book. 

It  is  not  difficult  to  prove  also  that  there  are  an  infinite  number 
of  primes  of  each  of  the  forms  6n —  1,  6n  -(-  1,  and  Sn  -f-  5.1 

These  are,  however,  only  special  cases  of  the  general  theorem 
that  in  every  unlimited  arithmetical  progression,  whose  general 
term  is  a  -f-  nd,  the  first  term  a  and  the  common  difference,  d, 
being  prime  to  each  other,  there  occur  infinitely  many  prime  num- 
bers. This  theorem  was  first  proved  by  Dirichlet  (see  D.  D.,  4th 
Ed.,  Sup.  VI),  but  he  did  not  give  an  interval  within  which  a  new 
prime  must  lie,  as  in  the  case  of  Euclid's  proof.  This  omission 
was  supplied  by  Kronecker  in  1885.     (See  above  reference.) 

Among  problems  relating  to  prime  numbers  which  still  await 
solution  is  first  of  all  that  known  as  the  problem  of  the  frequency 
of  the  primes.     It  consists  in  the  determination  of  the  number  of 

1  Kronecker:  Vorlesungen  iiber  Mathematik;  Part  II,  Vol.  I,  p.  438. 
Cahen:  Theorie  des  Nombres,  p.  318. 


12  THE   RATIONAL   REALM INTEGERS. 

positive  primes  less  than  any  given  positive  number  m,  that  is,  in 
the  determination  of  the  law  which  governs  the  distribution  of 
the  primes  among  the  entire  series  of  positive  integers. 

Kronecker  mentions  two  interesting  theorems  which  are  be- 
lieved to  be  true,  although  no  proofs  have  yet  been  obtained. 

I.  Every  positive  even  integer  can  be  represented  as  the  sum 
of  two  positive  prime  numbers  (2  excepted).  This  theorem  was 
first  stated  by  Goldbach,  then  by  Waring.  Kronecker  remarks1 
that  after  testing  this  theorem  for  the  even  integers  from  2  to 
2000,  it  is  observed  that  the  number  of  possible  representations 
of  2n  in  this  form  increases  as  n  increases,  which  heightens  the 
probability  of  correctness ;  for  example,  we  have 

4  =  2  +  2;  6  =  3  +  3;  8  =  3  +  55  10  =  3  +  7,  5  +  5; 
12  =  5+7;  14  =  3+11,  7  +  7',  16  =  3+13,  5+n; 
18  =  5  +  13,  7  +11;  etc. 

II.  Every  positive  even  integer  can  be  represented  in  infinitely 
many  ways  as  the  difference  of  two  positive  primes. 

If  the  truth  of  this  theorem  be  assumed  and  it  be  applied  to  the 
integer  2,  we  obtain  the  theorem :  However  far  we  may  go  in  the 
series  of  positive  primes,  we  shall  always  find  primes  which  differ 
only  by  2,  that  is,  which  lie  as  close  as  possible  together.  Natur- 
ally the  frequency  of  such  pairs  of  primes  decreases  the  farther 
out  we  go  in  the  series  of  positive  integers.  Among  the  first  one 
hundred  integers  there  are  eight  such  pairs : 

3,  5;  5,  7;  H*  13;  17,  19;  29,  3i;  4i,  43;  59,  61;  71,  73; 
and  among  the  second  hundred  seven : 

101, 103 ;  107, 109 ;  137, 139 ;  149, 151 ;  *79, 181 ;  191, 193 ;  197, 199. 
If  we  go  sufficiently  far  in  the  series  of  positive  integers  we  can 
find  as  great  a  number  of  successive  integers  as  we  please,  no 
one  of  which  is  a  prime,  for  none  of  the  integers  n!  +  2,  n!  +  3, 
-",nl-\-n  is  a  prime,  since  nl-^-i,  i^n,  is  divisible  by  *;  for 
example,  5  !  +  2,  5  !  +  3,  5  !  +  4,  5  !  +  5  are  all  composite  numbers. 

§7.     Unique  Factorization  Theorem. 

According  to  the  definition,  every  composite  number  can  be 
1  Vorlesungen  uber  Math.,  Part  II,  Vol.  I,  p.  68. 


THE   RATIONAL   REALM INTEGERS. 


13 


resolved  into  the  product  of  two  factors,  neither  of  which  is  a 
unit.  One  or  both  of  these  factors  may  be  composite,  and  hence 
in  turn  resolvable  into  two  factors,  neither  of  which  is  a  unit,  and 
we  can  continue  this  process  until  we  reach  factors  which  are 
primes.  It  is  evident  that  when  one  or  both  of  the  factors  are 
composite,  the  resolution  is  not  unique;  for  example,  210=14-15 
=  io-2i  =6-35  =  2- 105  =  3-70=  5-42  =  7.30.  We  shall  show 
that,  when  the  resolution  is  continued  until  the  factors  are  primes, 
it  will  be  unique,  considering  associated  factors  as  the  same  (see 
§  4) ,  and  that  such  a  resolution  is  always  possible ;  for  example, 


7 
7 
7 
7 
5; 


210=14-15   =2-7-3-5 
=  10-21   =2-5-3-7 
=  6-35   =2-3 
=  2-105  =  2-3- 

=  3-7o  =3-2 

=  5.42  =5-2. 

=  7-30  =7-2 

that  is,  210  can  be  represented  in  only  one  way  as  a  product  of 
prime  numbers. 

To  prove  this  theorem,  upon  which  the  whole  theory  of  the 
rational  integers  depends,  that  is,  that  every  rational  integer  can 
be  represented  in  one  and  only  one  way  as  a  product  of  prime 
numbers,  we  require  the  two  following  theorems : 

Theorem  A.  //  a  be  any  integer  and  b  any  integer  different 
from  0,  there  exists  an  integer  m  such  that 

\a  —  mb\<\b\ 


Let 


f^zm  +  r, 


where  01  is  the  integer  nearest  to  -  and  hence  |  r  |  ^  J ;  then  m  is 


the  required  integer,  for 


m 


<i, 


whence,  multiplying  by  |  b  |, 

\a  —  mb  I  <  I  6  |. 
This  theorem  is  equivalent  to  saying  that  we  can  divide  a  by  b 


rr  -me 


14  THE   RATIONAL   REALM INTEGERS. 

so  as  to  obtain  a  remainder  less  in  absolute  value  than  b,  the  quo- 
tient being  m.  There  are,  except  when  a  is  divisible  by  b,  evi- 
dently two  integers  which  satisfy  the  requirements  of  the  theorem, 
one  selected  as  above  and  another  differing  from  the  first  by  I ; 
for  example,  if  a  =12  and  b  =  — 5,  then 

|l2-(-2)(-s)|<|-5|   and   |i2_(_3)(_S)|<|_5|; 
and  hence  both  — 2  and  — 3  satisfy  the  requirements  of  the 
theorem,  -j-  2  being  the  integer  selected  as  in  the  proof. 

Theorem  B.  //  a  and  b  be  any  two  integers  prime  to  each 
other,  there  exist  two  integers,  x  and  y,  such  that 

ax  -{-by  =  1. 

If  either  a  or  b  be  a  unit,  the  existence  of  the  integers  x,  y^is 
evident.  We  shall  now  show  that,  if  neither  a  nor  b  be  a  unit, 
the  determination  of  x  and  y  can  be  made  to  depend  upon  the 
determination  of  a  corresponding  pair  of  integers  xx,  yx  for  a 
pair  of  integers  alf  bx  prime  to  each  other  and  such  that  one  of 
them  is  less  in  absolute  value  than  both  a  and  b. 

Assume  |&|<|a|,  which  evidently  does  not  limit  the  generality 
of  the  proof. 

By  Th.  A  there  exists  an  integer  m  such  that 
|a  —  mb\<\b\. 
Then  b  and  a  —  mb  are  a  pair  of  integers,  ax,  bx,  prime  to  each 
other,  and  a  —  mb  is  less  in  absolute  value  than  both  a  and  b. 

If  now  two  integers  xx,  yx  exist  such  that 

axxx  +  bxyx=fi; 

that  is,  bxx  +  (a  —  mb)yx=  1, 

we  have  ciyx-\-b(xx  —  myx)  =  i, 

and  hence  x=yti    y  =  x1  —  myx. 

The  determination  of  xx,  yx  for  ax,  bx  may,  if  neither  ax  nor  bx 
be  a  unit,  be  made  to  depend  similarly  upon  that  of  x2,  y2  for  a 
pair  of  integers  a2f  b2  prime  to  each  other  and  such  that  one  of 
them  is  less  in  absolute  value  than  both  ax  and  bx.  By  a  continua- 
tion of  this  process,  we  are  able  always  to  make  the  determination 
of  x  and  y  depend  eventually  upon  that  of  xn,  yn  for  a  pair  of 
integers  an,  b„,  one  of  which  is  a  unit. 


THE   RATIONAL   REALM INTEGERS.  I  5 

Since  the  existence  of  xn  and  yn  is  evident,  the  existence  of  x 
and  y  is  proved. 

Ex.     Let  a ■  =  14,  b  =  9 ;  then  oj  =  9,  b1  =  5,  and  the  determination  of 

x  and  ;y,  so  that                           14^  +  9y  =  1  2) 

depends  upon  the  determination  of  xlf  yu  so  that 

gxs  +  53>i  =  1.  3) 

We  can  make  the  determination  of  xu  yt  depend  upon  the  determination  of 
x2,  y2  for  the  pair  of  integers  a2  =  5,  b2  —  —  1,  but  it  is  sufficient  here  to 
notice  that  xx  xs  —  1,  yx  =  2  satisfy  3)  and  hence  ,r  =  y1  =  2,  y  =#i  —  m^ 
=  —  1  —  1  -2  =  —  3  satisfy  2) . 

The  problem  of  finding  the  two  integers  x  and  y  is  most  easily  solved 
by  the  method  of  continued  fractions,  but  the  form  of  proof  here  used 
to  show  the  existence  of  x  and  y  has  been  adopted  as  being  more  easily 
applicable  to  realms  of  higher  degree. 

The  proof  given  satisfies  completely,  however,  the  requirement  which 
Kronecker  considered  should  be  imposed  upon  every  existence  proof  in 
the  Theory  of  Numbers  (see  below)  ;  that  is,  it  furnishes  a  method  by 
which  in  a  finite  number  of  steps  the  desired  integers  x,  y  can  be  found 
from  the  given  ones  a,  b. 

Hensel  says  in  his  preface  to  Kronecker's  "Lectures  on  the  Theory  of 
Numbers,"  "  Kronecker  consciously  imposed  upon  the  definitions  and  proofs 
of  the  general  arithmetic  a  demand  whose  rigorous  observance  essentially 
distinguishes  his  exposition  of  the  theory  of  numbers  and  algebra  from 
almost  all  others. 

"  He  considered  that  one  can  and  must  so  formulate  each  definition  in 
this  domain  that  by  a  finite  number  of  trials  it  can  be  determined  whether 
or  not  it  is  applicable  to  any  proposed  quantity. 

"  Likewise  a  proof  of  the  existence  of  a  quantity  is  to  be  looked  upon 
as  rigorous  only  when  it  contains  at  the  same  time  a  method,  by  which 
the  quantity,  whose  existence  is  proved,  can  be  actually  found.  Kronecker 
was  very  far  from  throwing  entirely  aside  a  definition  or  proof  which  did 
not  satisfy  these  high  requirements,  but  he  considered  that  something 
was  still  wanting  and  he  held  its  completion  in  this  direction  to  be  an 
important  task,  by  which  our  knowledge  would  be  extended  in  an  es- 
sential point." 

"  He  considered,  moreover,  that  a  formulation  rigorous  in  this  sense 
was  in  general  of  simpler  form  than  another  which  did  not  satisfy  this 
demand  and  he  has  in  many  cases  shown  by  his  lectures  that  this  is 
the  case." 

Cor.  //  a  and  b  be  any  two  rational  integers,  there  exists  a 
common  divisor  d  of  a  and  b  such  that  every  common  divisor 


1 6  THE   RATIONAL   REALM INTEGERS. 

of  a  and  b  divides  d,  and  we  can  find  two  integers  x  and  y  such 

that  ax  -\-by  =  d. 

Let  a  =  axc,  b  =  bxc, 

where  ax  and  bx  and  prime  to  each  other. 

By  Theorem  B  two  integers  x  and  y  exist  such  that 

axx  +  bxy  =  i.  i) 

Multiplying  I )  by  c,  we  have 

axcx  +  bxcy  =  c ; 

that  is  ax  -\-by  =  c. 

Every  common  divisor  of  a  and  b  evidently  divides  c.  Hence 
c  is  the  divisor,  d,  sought. 

We  call  d  the  greatest  common  divisor  of  a  and  b. 

It  is  evident  that  two  such  divisors  which  are  not  associates 
cannot  exist;  for  if  dx,  d2  be  two  such  divisors,  then  since  from 
the  definition  dx  must  divide  d2  and  d2  must  divide  dlt  dx  and  d2 
are  associates. 

Any  number  of  integers,  alt  a2,  •••,  an,  possess  a  common  di- 
visor which  is  divisible  by  all  common  divisors  of  these  integers ; 
for  let  dx  be  the  greatest  common  divisor  of  alf  a2  as  defined 
above.     Then  two  integers,  xx  and  x2,  exist  such  that 

a\xx  +  a2x2  —  ^1- 
Let  now  d2  be  the  greatest  common  divisor  of  dx  and  a3.     It  is 
evident  that  d2  is  a  common  divisor  of  ax,  a2,  a3,  and  that  two 
integers,  yx,  y2,  exist  such  that 

d1y1  +  a3y2  =  d2, 

or  axxxyx  +  a2x2yx  +azy2  =  d2 ; 

that  is,  three  integers,  zx,  z2,  zz,  exist  such  that 

axzx  +  a2z2  +  a3z3  =  d2, 

from  which  identity  it  is  evident  that  every  common  divisor  of 
ax>  a2}  a3,  divides  d2. 

Proceeding  similarly  with  d2  and  a4,  then  with  their  greatest 


THE   RATIONAL   REALM INTEGERS.  1 7 

common  divisor  d3  and  a5,  etc.,  we  see  finally  that  there  exist  n 
integers  ulf  u2,  •-,un  such  that 

aiMi  +  °2 u2  ~f~  '  * "  ~f"  flnWn  =  d, 

where  d  is  a  common  divisor  of  ax,  a2,  •  •  *,  an. 

From  this  identity  it  is  evident  that  every  common  divisor  of 
a1}  a2,  ...  a-n  divides  d.  We  call  d  therefore  the  greatest  common 
divisor  of  the  n  integers  ax,  a2y  •  •  •,  an. 

The  common  divisors  of  a  system  of  integers  are  evidently  the 
divisors  of  the  greatest  common  divisor  of  the  system. 

To  find  the  greatest  common  divisor  of  n  integers  a1}  a2,  •  •  •,  o«, 
we  find  the  greatest  common  divisor  dx  of  ax  and  a2 ;  then  the 
greatest  common  divisor  of  dx  and  az,  which  will  evidently  be  the 
greatest  common  divisor  of  alt  a2,  a3. 

Proceeding  in  this  manner  we  arrive  finally  at  an  integer  d 
which  is  the  greatest  common  divisor  of  all  of  the  integers.  In 
particular,  if  a^,  a2,  •••,  a„  have  the  greatest  common  divisor  I, 
we  have 

a1u1  +  a2u2  +  •  *  *  +  &nUn  =  i. 

This  corollary  is  usually  known  as  the  greatest  common  divisor 
theorem  and  can  be  proved  independently  of  Theorem  B  which 
follows  easily  from  it. 

The  independent  proof  of  the  corollary  depends  upon  Theorem  A  and 
the  following  simple  theorem  whose  truth  is  obvious. 

If  a  =  mb  +  r,  then  every  integer  which  divides  both  a  and  b  divides 
both  b  and  r,  and  vice  versa;  that  is,  the  common  divisors  of  a  and  b 
are  identical  with  the  common  divisors  of  b  and  r. 

By  virtue  of  these  two  theorems  we  are  able  to  substitute  for  the 
problem  of  finding  the  integer  which  is  divisible  by  all  common  divisors 
of  a  and  b  (|&|  =  |fl|)  the  corresponding  problem  for  the  two  integers 
b  and  r,  where  a  =  mb-\-r,  and  |  r  \  <  |  b  |.* 

From  Theorem  A,  it  is  evident  that  we  can  form  a  chain  of  identities, 

a  =  mb  -f-  r, 
b  =  mjr  -f  n, 
r  =  m2rt  +  r2, 

1  Euclid :  Elements,  Book  VII,  Prop.  2. 

2 


I  8  THE   RATIONAL   REALM INTEGERS. 

in  which  |  r  |  >  |  rx  \  >  \r2\,  etc.,  arriving  after  a  finite  number  of  such 
steps,  since  the  integers  less  in  absolute  value  than  a  given  integer  are 
finite  in  number,  at  a  remainder  rn+i  which  is  o,  and  hence 

rn-i  =  mn+1rn 

Now  from  the  theorem  above  it  is  evident  that  the  common  divisors  of 
a  and  b  are  identical  with  the  common  divisors  of  b  and  r,  and  hence  with 
those  of  r  and  n,  and  finally  with  those  of  rni  and  rn. 

But  rn  is  a  common  divisor  of  r%^  and  rn  and  evidently  is  divisible  by 
every  common  divisor  of  rn-x  and  r«.  Hence  rn  is  the  desired  common 
divisor  of  a  and  b ;  that  is,  it  is  divisible  by  all  the  common  divisors  of  a  and 
b.  Moreover,  we  can  by  means  of  the  method  of  continued  fractions  ex- 
press d,=rn,  in  the  form 

ax  +  by  =  d.1 

The  greatest  common  divisor  of  two  or  more  integers  is  seen 
to  be  the  common  divisor  of  greatest  absolute  value,  there  being 
only  one  such  common  divisor  since,  if  |  a  |  =  |  b  |,  then  a  and.  b 
are  associates.  It  is  also,  as  we  have  seen  from  the  proof  of  the 
above  corollary,  the  common  divisor  such  that  the  quotients  ob- 
tained by  dividing  each  of  the  integers  by  this  divisor  have  no 
common  divisor  other  than  ±  I. 

The  reason  why  neither  of  these  properties  has  been  chosen 
for  the  definition  of  the  greatest  common  divisor  of  two  or  more 
integers  will  appear  later  (see  p.  252). 

An  objection  to  the  former  of  the  two,  which  is  the  one  usually 
employed  is,  however,  immediately  evident  in  that  the  idea  of 
inequality  is  introduced,  whereas  the  question  is  purely  one  of 
divisibility. 

Theorem  C.  //  the  product  of  two  integers,  a  and  b,  be  divis- 
ible by  a  prime  number,  p,  at  least  one  of  the  integers  is  divisible 
by  p. 

Let  ab  —  cp,  and  assume  a  not  divisible  by  p.  Then  a  and  p 
have  no  common  divisor,  and  there  exist  two  integers,  x  and  y, 

such  that  ax  -\-  py  =  1.  1 ) 

xCahen:  p.  60.  Bachman :  Niedere  Zahlentheorie,  p.  107.  Chrystal : 
Vol.  II,  p.  445. 


THE   RATIONAL    REALM INTEGERS.  1 9 

Multiplying  I )  by  b,  we  have 

bax  +  bpy  =  b, 
and  therefore  (ex  +  by)p  =  b, 

where  ex  -\-  by  is  an  integer.     Hence  b  is  divisible  by  p. 

Cor.  1.  //  the  product  of  any  number  of  integers  be  divisible 
by  a  prime  number,  p,  at  least  one  of  the  integers  is  divisible  by  p. 

Cor.  2.  //  neither  of  two  integers  be  divisible  by  a  prime  num- 
ber, p,  their  product  is  not  divisible  by  p. 

Cor.  3.  //  the  product  of  two  integers,  a  and  b,  be  divisible 
by  "an  integer  c  and  neither  a  nor  b  be  divisible  by  c,  then  c  is  a 
composite  number. 

Theorem  i.  Every  rational  integer  can  be  represented  in  one 
and  only  one  way  as  the  product  of  prime  numbers. 

Let  m  be  a  rational  integer.  If  m  be  a  prime,  the  theorem  is 
evident.  Let  m  be  a  composite  number  ;  m  then  has  some  divisor, 
a,  other  than  ±mor  ±  i.  Either  a  is  a  prime  or  it  has  some 
divisor,  b,  other  than  ±oor±i.  If  &  be  not  prime,  it  has  some 
divisor,  c,  other  than  ±  i  and  ±  b.  Proceeding  in  this  manner, 
we  must  at  last  arrive  at  a  prime  number,  for  the  integers  of  the 
series  a,  b,  c,  ••  •,  decrease  in  absolute  value,  and  since  there  are 
only  a  finite  number  of  integers  smaller  in  absolute  value  than 
m,  the  series  can  have  only  a  finite  number  of  terms,  the  last  of 
which  will  be  a  prime  number ;  for  otherwise  the  series  could  be 
extended.  Let  this  prime  be  px.  By  §3,  I,  px  is  a  factor  of  m 
and  we  have  m  =  p1m1.  If  mx  be  a  prime,  the  resolution  of  m 
into  its  prime  factors  is  complete.  If  ;;zx  be  a  composite  number, 
it  contains  a  prime  factor,  p2,  and  we  have 

m1  =  p2m2, 

or  m  =  p1p2m2. 

If  m2  be  not  a  prime,  we  can  proceed  as  before  until  we  have 
resolved  m  into  factors',  all  of  which  are  primes.  That  there  will 
be  only  a  finite  number  of  these  factors  is  evident  from  the  fact 


20  THE   RATIONAL   REALM INTEGERS. 

that  the  integers  of  the  series,  m,  m1}  m2,  •  •  •,  decrease  in  absolute 
value  and  hence  must  be  finite  in  number. 

We  have  now  shown  that  the  representation  of  an  integer  as  a 
product  of  a  finite  number  of  primes  is  always  possible.  It  re- 
mains to  be  proved  that  this  representation  is  unique,  regarding 
representations  as  identical,  which  differ  only  by  the  substitution 
for  a  prime  of  its  associate. 

Let  m  =  pxp2pz  --pr  =  q1q2qs  ~-q, 

be  two  representations  of  m  as  a  product  of  prime  numbers. 

Since  the  product  q1q2"-q8  is  divisible  by  plt  at  least  one  of 
its  factors,  say  qlf  must  be  divisible  by  px.  But  qx  has  only  the 
divisors  ±  qx  and  ±  i.  Hence  q1=±p1;  that  is,  qx  is  asso- 
ciated with  pv     Then  follows 

Pip*"'  pr=±q2qz---  q*- 

In  the  same  manner  we  can  show  that  some  factor  of  the  product 
q<&z "*<7«  is  associated  with  p2,  and  proceeding  similarly  we  can 
show  that  for  each  prime  that  occurs  once  or  oftener  as  a  factor 
of  the  product,  pxp2pz  ■  •  •  pr,  there  occurs  at  least  as  often  an  asso- 
ciated prime  in  the  product  qxq2qz  •  •  ■  q8-  In  like  manner,  we  can 
show  that  for  each  prime  which  occurs  once  or  oftener  as  a  factor 
of  the  product  qxq2q3  •••#«,  there  occurs  at  least  as  often  an  asso- 
ciated prime  in  the  product  pxp2pz  •"  pr>  Hence  the  two  repre- 
sentations are  identical.  We  can  simplify  the  representation  of  a 
composite  number  as  the  product  of  its  prime  factors  by  express- 
ing the  product  of  associated  prime  factors  as  a  power  of  one  of 
them.  Thus,  if  of  the  prime  factors  of  m,  ex  are  associated  with 
plf  e2  with  p2, -",er  with  pr,  we  can  write 


m  =  ±  p!eip2e2 '"  P 


Cr 

r     • 


Cor.  i.  If  a  and  b  be  prime  to  each  other  and  c  be  divisible  by 
both  a  and  b,  then  c  is  divisible  by  their  product. 

Cor.  2.     //  a  and  b  be  each  prime  to  c,  then  ab  is  prime  to  c. 

Cor.  3.  //  a  be  prime  to  c  and  ab  be  divisible  by  c,  b  is  divis- 
ible by  c. 


THE   RATIONAL   REALM INTEGERS.  21 

Theorem  2.    If 

U  (*)  =  a^  +  a^m~x  +  •  -  •  +  a™> 

f2  (x)  =  &0^»  +  bxX^  + .  • .  +  bn, 

be  any  two  integral  functions  of  x,  whose  coefficients  are  rational 
integers,  having  in  each  case  no  common  divisor,  then  the  coeffi- 
cients of  the  product  of  these  functions 

are  rational  integers  without  a  common  divisor. 

If  the  coefficients  c0,  cx,  •  •  •,  cm+n  of  f(x)  have  a  common  divisor 
other  than  ±  I,  there  must  be  at  least  one  prime  number  which 
divides  all  of  them. 

Let  p  be  such  a  prime  and  suppose  that  p  divides 

cto,  ax,  •  -  -,  ar_!,  but  not  ar, 

and  b0,  blt  •  •  •,  b8_x,  but  not  b8y 

where  in  accordance  with  our  original  assumption  that  the  coeffi- 
cients of  fx(x)  and  f2(x)  have  no  common  divisors, 

o^rgfw  and  O&sgn. 

We  have  now 

cr+8  =  arb8  +  ar_xb8+x  +  Or.2b8+2  +  '  • '  +  ar+ib8_x  +  ar+2b8_2  +  •  •  •. 

It  is  evident  that  cr+8  is  not  divisible  by  p,  for  arb8  is  not  divisible 
by  p,  neither  ar  nor  b8  being  divisible  by  p,  while  all  the  remain- 
ing terms  are  divisible  by  p,  since  each  of  these  terms  contains  as 
a  factor  some  one  of  the  coefficients  ao,ax,~',ar-1,b0,b1,---,b8_1, 
which  are  all  divisible  by  p. 

Hence  the  coefficients  of  f(x)  have  no  common  divisor. 

Theorem  3.    // 

fx  O)  ==  *•  +  axx^  +  . . .  +  am, 

f2(x)=x"  +  bxx^  +  --  +  bn 

be  two  rational  integral  functions  of  x,  the  coefficients  of  the 


22  THE    RATIONAL   REALM INTEGERS. 

highest  powers  of  x  in  each  case  being  i,  and  the  remaining  coeffi- 
cients rational  numbers,  the  coefficients,  clt  c2,  •  •  •,  cm+n  of  their 
product 

f(x)  =/, (x)  •  f2(x)  =x~*  +  cxx***-*  +  •  •  •  +  cm+n 

cannot  all  be  rational  integers  unless  all  of  the  coefficients  alf  a2,  •••, 
am,  blt  b2,  '",bn  are  rational  integers.1 

Let  a0  and  b0  be  the  least  common  denominators  of  the  coeffi- 
cients of  ft(x)  and  f2(x)  respectively.  Then  each  of  the  func- 
tions a0f1(x)  and  b0f2(x)  has  rational  integral  coefficients  without 
a  common  divisor.  If  now  the  coefficients  clt  c2,  •  •  •,  cm+n  are  to 
be  integers,  the  coefficients  of  the  product, 

a<A>AO)  •  /.(*)  =«A/(*), 
must  all  be  divisible  by  a0b0. 

But  by  Th.  2  this  is  impossible  unless  a0=i,  b0  =  i ;  that  is, 
ax,  a2,  •  •  • ,  am,  b0,  bx,  •  •  •,  bn  are  integers. 

Theorem  4.  A  necessary  as  well  as  sufficient  condition  that  an 
algebraic  number  a  shall  be  an  algebraic  integer  is  that  the  coeffi- 
cients of  the  single  rational  equation  of  lowest  degree  of  the  form 

fx(x)  =xl  +a1xl~1  +  •••  +  at  =  o,  1) 

ivhich  it  satisfies,  shall  be  rational  integers. 
If  a  satisfy  an  equation 

f2(x)  =  xm  +  M*w_1  +  '  • '  +  bm  =  o, 

of  degree  higher  than  the  /th  whose  coefficients  are  rational  num- 
bers, then  by  Chap.  I,  Th.  1, 

where  f3(x)  is  a  rational  integral  function  of  x  with  rational 
coefficients,  the  coefficient  of  its  term  of  highest  degree  being  1. 
But  by  Th.  3  the  coefficients  of  f2(x)  cannot  all  be  rational  in- 
tegers unless  the  coefficients  of  fx(x)  are  all  rational  integers. 
Hence  the  theorem. 

1  Gauss :  Disq.  Arith.,  Art.  42,  Works,  Vol.  I. 


THE   RATIONAL    REALM INTEGERS.  23 

We  see  therefore  that  the  system  of  rational  integers  and  that 
of  the  integers  of  R  are  coextensive,  and  hence  that  all  that  has 
been  said  in  the  preceding  pages  concerning  rational  integers  may 
now  be  looked  upon  as  applying  to  the  integers  of  R.  Hereafter 
the  terms  rational  integers  and  integers  of  R  will  be  used  inter- 
changeably. 

It  is  seen  from  the  above  theorem  that  the  equation  of  lowest 

degree  6i  the  form  i)  satisfied  by  an  algebraic  number,  determines 

not  only  the  degree  of  the  number,  but  whether  it  is  or  is  not  an 

algebraic  integer. 

After  having  proved  the  unique  factorization  theorem  we  could  have 
shown  that  no  rational  fraction  alb,  where  a  and  b  are  prime  to  each 
other  and  fr=j=±  i,  can  satisfy  an  equation  of  the  form  i)  whose  coef- 
ficients are  rational  integers  and  hence  that  the  only  integers  of  R  are 
the  rational  integers,  but  it  has  seemed  better  to  treat  the  question  in 
the  general  manner  we  have  used  above. 

§  8.    Divisors  of  an  Integer. 

We  can  now  exhibit  in  a  very  convenient  form  all  divisors  of 
any  given  integer,  m,  and  deduce  therefrom  simple  expressions 
for  the  number  and  the  sum  of  these  divisors.  Let  m  be  written 
in  the  form 

m  =  ±  pxeip2e2 ' ' '  prer, 

where  plf  p2,  '-,pr  are  the  different  prime  factors  of  m. 

If  d  be  a  divisor  of  m,  it  can  contain  as  factors  only  those 
primes  which  occur  in  rn,  but  each  of  these  primes  can  occur  in  d 
to  any  power  not  greater  than  that  to  which  it  occurs  in  m;  that 
is,  every  divisor  of  m  must  have  the  form 

d=±pimipm2...  prmr} 

where  o^mig^.;  %  =5=  1, 2,  ••  *, r, 

and  each  of  the  integers  obtained  by  giving  these  different  values 
to  mlfm2f  "',mr  is  a  divisor  of  m.  We  can  now  easily  obtain  an 
expression  for  the  number,  AT,  of  the  different  divisors  of  m, 
associated  divisors  being  considered  as  identical.  Since  there  are 
e1  -f- 1,  e2-\-i,  •  ••,  er-\-i  possible  values  for  m1,  m2,  •••,  mr 
respectively,  there  are  (^i  +  i)(^2+I)""(^r+i)  different  sets 
of  values  of  mlf  m2,  •••,  mr  and  each  of  these  sets  gives  a  dif- 


24  THE   RATIONAL   REALM INTEGERS. 

ferent  divisor  of  m.     Moreover,  these  sets  of  values  of  m19  m2,  •  •  •, 
mr  give  all  the  different  divisors  of  m,  whence  we  have 

N=(e1+i)(e2  +  i)-.-(er  +  i). 

We  can  find  similarly  an  expression  for  the  sum,  S,  of  the  dif- 
ferent positive  divisors  of  m. 
On  expanding  the  product 

'~(I+Pr  +  Pr2+---Pr"), 

we  obtain  a  series,  all  of  whose  terms  are  positive  divisors  of  m, 
each  positive  divisor  of  m  occurring  once  and  but  once.     The  sum 
of  this  series  is  therefore  S. 
Hence 

S=(i+p1  +  p12-i---'+p^)(i+p2  +  p22+"-p^) 

~'(I+Pr  +  Pr2+--Prer) 

_Aei+1  -  i  .  A*+1  -  J . . .  Aer+1  -  * 
A    -  i'   '    A-  1  A-  i 

Ex.    Let  m  =  6o  =  22-3-5. 

We  have  #=  (2+  1)  (1  +  1)  (1  +  1)  —  12, 

and  Saz^ii  ^ZZl .  5jhI=7.4.6  _  l68 

2—1     3  —  1     5  —  1 

results    which    are    easily    substantiated    bv    observing   that    the    positive 
divisors  of  60  are  J,  2,  3,  4,  5,  6,  10,  12,  ic.  20,  30  and  60. 

We  observe  that  N  depends  only  upon  the  exponents  of  the 
powers  to  which  the  different  prime  factors  appear  in  m,  while  S 
depends  also  upon  the  absolute  values  of  these  primes. 

We  have  defined  (§3)  a  common  divisor  and  a  common  mul- 
tiple of  two  or  more  integers.  The  representation  of  an  integer 
as  a  product  of  its  different  prime  factors  leads  us  to  convenient 
expressions  for  the  common  divisors  and  common  multiples  of 
a  system  of  integers. 

Let  m^,  Wv,,  •••,mic  be  any  system  of  integers  and  suppose  each 
integer  of  this  system  expressed  as  a  product  of  powers  of  its 
different  prime  factors.     Let  plt  p2,  •  •  •,  pr  be  the  different  prime 


THE   RATIONAL   REALM INTEGERS.  2$ 

factors  of  mx,m2,  --',^1^',  lx,l2,  --,lr,  the  exponents  of  the  lowest 
powers,  and  glt  g2,  •  •  •,  gr,  the  exponents  of  the  highest  powers  to 
which  plt  p2,  -",pr  occur  in  any  of  these  integers.  Remembering 
now  that  every  common  divisor  of  mlt  m2,  •  •  •,  m*,  can  contain  as  a 
factor  a  prime,  pi,  to  a  power  not  higher  than  the  lowest  to  which 
pi  occurs  in  any  of  the  integers  mx,m2,  ~',mk,  we  see  that  every 
common  divisor  of  mx,  m2,  •  •  •,  wfc,  has  the  form 

where  o^di^h;  i=i,2,--,r. 

When  dltd2t  '"$dr  have  their  greatest  possible  values,  that  is, 
Kt  h> ' '  '$  k,  the  divisor  so  obtained,  is  evidently  the  greatest  com- 
mon divisor  of  «»!,%•",%  Denoting  the  greatest  common 
divisor  of  mx,m2,  ~',mu,  by  g,  we  have  therefore 

g  =  PihP2l2--prlr. 

Likewise  since  every  common  multiple  of  m^,  m2,  •  •  •,  mk,  must 
contain  as  a  factor  a  prime,  pi,  at  least  to  the  highest  power  to 
which  pi  occurs  in  any  one  of  the  integers  %  m2,  •  •  •,  m*,  we  see 
that  every  common  multiple  of  mlf  m2,  •  •  •,  m*  has  the  form 

apSW---  prnr, 
where  **$gi,  i=i,2,--  -,  r, 

and  a  is  any  integer. 

When  nx,n2,---,nr  have  their  least  possible  values,  that  is, 
gi>g2>'">gr,  and  a  is  a  unit,  the  multiple  obtained  is  the  least 
common  multiple  of  mlt  m*>,  •  *  •,  m*.  Denoting  the  least  common 
multiple  of  mx,  m2,  -•-,  m*  by  I,  we  have  therefore 

l  =  p^p29*---pr9r. 

We  observe  that  just  as  the  common  divisors  of  a  system  of  in- 
tegers are  the  divisors  of  the  greatest  common  divisor  of  the  sys- 
tem, so  every  common  multiple  of  all  the  integers  of  the  system 
is  a  multiple  of  their  least  common  multiple.  When  two  or  more 
of  the  integers  mlt  m2,  •  •  •,  mjc  are  prime -to  each  other,  the  greatest 
common  divisor  of  the  system  is  evidently  a  unit,  and  when  the 
integers  mx,  m2,  •  ■  •,  m&  are  prime  each  to  each  their  least  common 


26  THE   RATIONAL   REALM INTEGERS. 

multiple  is  their  product,  mxm2  •  •  •  mk.     If  an  integer  be  divisible 
by  each  one  of  a  system  of  integers  mlf  m2>  •  ■  •,  w&,  it  is  divisible 
by  their  least  common  multiple. 
If  we  have  two  integers 

0  =  Px*Pf*'  '  •  pra%   b  =  p^pj*  •  •  •  />r6r, 

and  g ■  =  p1hp2h  ...  prirf  l=pjhpp...  prffr 

be  respectively  their  greatest  common  divisor  and  least  common 
multiple,  it  is  evident  that 

li  +  gi  =  <h  +  K  h  +  g2  =  ^  +  b2r"Jr  +gr  =  ar  +  br, 
and  hence  that  gl=  ab  ;  that  is,  the  product  of  two  integers  is  equal 
to  the  product  of  their  greatest  common  divisor  and  least  common 
multiple;  for  example 

12 -30  =  6-60. 

The  representation  of  an  integer  m  as  a  product  of  powers  of 
its  different  prime  factors  gives  us  also  a  criterion  for  determin- 
ing whether  in  is  or  is  not  the  &th  power  of  an  integer. 

Let  m  =  ±  pxeip2e2 ' ' '  prer. 

By  putting  m  =  nk,  we  see  immediately  that  the  necessary  and 
sufficient  condition  that  m  shall  be  the  &th  power  of  an  integer  is, 
if  k  be  odd,  that  elf  e2,--,er  shall  be  divisible  by  k,  while  if  k  be 
even  there  is  the  further  condition  that  m  shall  be  positive. 

§  9.  Determination  of  the  Highest  Power  of  a  Prime,  p,  by 
which  m !  is  divisible. 

The  method  employed  consists  in  counting,  successively,  those 
:nte£ers  of  this  product  which  are  divisible  by  p,  p2,  p3,  etc., 
respectively.  Remembering  that  those  integers  which  are  divis- 
ible by  pl  have  already  been  counted  i —  1  times,  as  among  those 
divisible  by  p,  p2,  •••,  pl~x,  the  sum  of  these  enumerations  is  seen 
to  be  the  exponent  of  thi  desired  power  of  p.  Denote  this  expo- 
nent by  e.  Since  e  will  have  the  same  value  for  —  p  as  for  p,  we 
can  without  loss  of  generality  assume  p  positive. 

Let  [a/b]  denote  the  greatest  integer  contained  in  the  fraction 
a/b,  where  a  and  b  are  both  positive;  in  particular   [a/b]   is  o 


THE  RATIONAL   REALM — INTEGERS.  2 J 

when  a  is  less  than  b.  Put  [m/p]  =  mlf  [m/p2]  =  m2,  •  •  •,  [m/p*] 
=«*,  •  •  • .     There  are  in  the  product 

i»/=i*2'3  •••  m, 
the  m,  integers,  />,  2/>,  Zp,-"^xP,  1 1 

divisible  by  /^  and  wz/  is  therefore  certainly  divisible  by  ^"i; 
that  is,  e  <  wx. 

In  like  manner  there  are  in  ml  the  m2  integers 

p\2p2,--;m2p2  2) 

divisible  by  p2.  We  have  counted  these  integers  once  already 
among  the  integers  i),  but  since  they  each  contain  p  twice  as  a 
factor,  and  there  are  m2  of  them,  we  must  add  m2  to  the  exponent 
of  the  power  of  p  which  is  known  to  divide  ml.  Hence  ml  is 
certainly  divisible  by  pm^m* ;  that  is, 

e  <£  mi  +  m2- 
Likewise  there  m3  integers  of  ml  divisible  by  p3,  each  of  which 
has  been  counted  twice  already.     Hence 

e  <  *>h  +  ™2  +  "h- 

Continuing  this  process  we  arrive  finally  at  a  fraction  m/pk>  which 
is  less  than  i,  and  hence 


<*-[$] -a 


The  highest  power  of  p  by  which  ml  is  divisible  is  therefore 

pmiHn2+--+mk-1}     whose     exponent     e     is      [m/p]  +  [m/p2]  -| 

+  [*•/£***]. 

If  p  >  m,  then  w1  =  o,  and  hence  e  =  o. 

Ex.     Let  m  =  ioo,  and  p  =  .3 ;  then 

W,=  [W]=  3, 
«4=[W]=  I, 
w»=[Hf]=  o, 

and  ^  =  33  +  n+3-J-i  =  48. 

It  is  easily  shown  that 

L;l 


m- 


28  THE   RATIONAL  REALM INTEGERS. 


and  hence 


-M 


Using  this  fact  in  the  example  just  given  we  have  Wi=  [x§-]  =  33>  m* 
=  [¥]  =  ii,  w3  =  [-V-1  =3,  ™*=  [f ]  =  Xj  «»=  CH  =  o. 


m/ 


§  io.    The  Quotient      7         ,  ,  where  m  =  a-\-b  -\ +  fc,  is 

a/&/  •••  &/ 
an  Integer.1 

This  quotient  will  be  recognized  as  the  so-called  multinomial 
coefficient ;  that  is,  the  coefficient  of  xxax*  •  •  •  xrk  in  the  expansion 
of  (x1  +  x2  -f-  •  •  •  xr) m.  When  r  =  2,  and  m  =  a-\-b,  we  have 
the  binomial  coefficient ;  that  is,  the  coefficient  of  xxax2m-a  in  the 
expansion  of  {xx-\-  x2)m. 

This  theorem  is  easily  proved  by  means  of  that  of  the  last  sec- 

tion.     To  show  that  .  T      '    .  ,  i) 

is  an  integer  it  is  necessary  and  sufficient  to  show  that  every 
prime,  p,  is  contained  to  as  high  a  power  in  the  numerator  as  in 
the  denominator.  Let  e,  ax,  blt  ■  •  •,  klt  be  the  exponents  of  the 
highest  powers  to  which  p  is  contained  in  ml,  al,  b!,'-,k!,  respect- 
ively.    We  must  show  that 

**«k  4- &!  +  "•■+"** 

Since  m= a  +  b  -f-  *  •  *  +  k, 

.                                  mad  k 

it  follows  that  —  =  -  H 1 4-  — , 

-«-  [7]i4+L-]*--[i] 

[?3*[?]*&+~[?} 


;;/ 


The  truth  of  this  theorem  is  at  once  evident  since  —m ; -    is  the 

alol  •  •  •  k! 

number  of  permutations  of  m  things  a,  b,  •  • ;  k  of  which  are  alike. 


THE   RATIONAL   REALM INTEGERS.  20, 

Hence,  by  addition, 

r  m~\ 

+  ••• 


[7H7V  Aj] 

*]+[f]+  +L:] 


+[f]+[?]+  +[?] 

•■-[i]+[?i+-  +[?]+- 


Hence  £  ^  Oj_  +  &x  +  •  •  •  +  kx. 

Therefore  p  is  contained  to  at  least  as  high  a  power  in  the 
numerator  of  I )  as  in  the  denominator.  But  p  was  any  prime ; 
therefore  I )  is  an  integer. 

From  this  it  follows  that  the  product  of  any  m  successive  posi- 
tive integers  is  divisible  by  ml 
For 

(a-f-i)  (g+2)' '  •(fl+ftQ  _  a/(a+i)(a+2)-"(o+w)  _  (g+w)' 
ml  aim!  at  ml 

which  is  an  integer.  From  this  and  the  fact  that  o  is  included 
among  m  successive  integers  which  are  not  all  positive  or  all  neg- 
ative, it  follows  that  the  product  of  any  m  successive  integers  is 
divisible  by  ml 


30  the  rational  realm integers. 

Examples.1 
'  i.  The  sum  of  two  odd  squares  can  not  be  a  square. 
,   2.  Every  integer  of  the  form  4» — I  has  an  odd  number  of 
factors  of  the  form  4ft —  1. 

3.  Every  prime  greater  than  5  has  the  form  yym'-^n  where 
w=i,  7,  11,  13,  17,  19,  23  or  29J 

4.  The  square  of  every  prime  greater  than  3  is  of  the  form 
24m  +  1,  and  the  square  of  every  integer  which  is  not  divisible  by 
2  or  3  is  of  the  same  form. 

5.  If  n  differ  from  the  two  successive  squares  between  which  it 
lies  by  x  and  y  respectively,  prove  that  n  —  xy  is  a  square. 

6.  The  cube  of  every  rational  integer  is  the  difference  of  the 
squares  of  two  rational  integers. 

7.  Any  uneven  cube,  n3,  is  the  sum  of  n  consecutive  uneven 
integers,  of  which  n2  is  the  middle  one. 

8.  Show  that  x3  —  x  is  divisible  by  6  if  x  be  any  integer. 

9.  Show  that  x4  —  4X3  +  Sx2  — 2X  1S  divisible  by  12  if  x  be 
any  integer. 

{o.  Show  that  x4m  -f-  x2m  +  1  never  represents  a  prime  number 
if  x  be  any  integer  other  than  1. 

1 1. r  Show  that  (mn)  !  is  divisible  by  (m!)nn! 

12.  Show  that  (2w)  !{2n) !  is  divisible  by  ml  nl  (m  +  m)  / 

13.  What  is  the  least  multiplier  that  will  convert  945  into  a 
complete  square? 

14.  Find  the  number  of  the  divisors  of  2160  and  their  sum. 

15.  Find  a  number  of  the  form  2n'3-a  (a  being  prime)  which 
shall  be  equal  to  half  the  sum  of  its  divisors  (itself  excluded). 

1  See  Chrystal;  Algebra,  Part  II,  pp.  506,  518  and  526  for  other  examples, 
also  C.  Smith,  Algebra,  and  Hall  and  Knight,  Higher  Algebra. 


CHAPTER   III. 
The  Rational  Realm. 

congruences. 

§  i.     Definition.     Elementary  Theorems. 

//  the  difference  of  two  integers,  a  and  b,  be  divisible  by  an 
integer  m,  a  and  b  are  said  to  be  congruent  to  each  other  with 
respect  to  the  modulus  m.     This  relation  is  expressed  by  writing 

a^=b,  modm.1 

Similarly,  if  the  difference  of  a  and  b  be  not  divisible  by  m,  we 
say  that  a  and  b  are  incongruent  to  each  other,  with  respect  to 
the  modulus  m,  and  write 

a^=b,  mod  m. 

Ex.  We  say  that  21  is  congruent  to  15  with  respect  to  the  modulus  3, 
since  21  — 15  is  divisible  by  3.  In  the  above  notation  this  fact  is  ex- 
pressed by  writing  21  ==  15,  mod  3. 

We  can  express  the  fact  that  a  is  congruent  to  b  by  writirlg 

a  —  b  =  km,  or  a  =  b  +  km, 

where  k  is  an  integer,  but  the  notation  a==b,  mod  m,  which  is  due 
to  Gauss,  has  the  great  advantage  of  placing  in  evidence  the 
analogy  between  congruences  and  equations ;  and  we  shall  see 
that  most  of  the  transformations  to  which  equations  can  be  sub- 
jected are  also  applicable  to  congruences. 

H.  J.  S.  Smith  says :  "  It  will  be  seen  that  the  definition  of  a 
congruence  involves  only  one  of  the  most  elementary  arithmetical 
conceptions, — that  of  the  divisibility  of  one  number  by  another. 
But  it  expresses  that  conception  in  a  form  so  suggestive  of  anal- 
ogies with  other  parts  of  analysis,  so  easily  available  in  calcula- 
tion and  so  fertile  in  new  results  that  its  introduction  into  arith- 

1  The  author  has  adopted  a  slight  variation  of  Gauss's  notation, 
a  =  b  (mod   m),  due,  he  believes,  to  H.  J.  S.  Smith. 

31 


32  THE   RATIONAL   REALM INTEGERS. 

metic  (by  Gauss)  has  proved  a  most  important  contribution  to 
the  progress  of  the  science." 

We  have  as  direct  consequences  of  the  de'finition  of  congruences 
the  following: 

i.  If  a  =  b,  modw,  i) 

and  b  =  c,  modw/  2) 

then  a  =  c,  modm; 

for,  from  1)  and  2),  we  have  respectively 

a  —  b  =  km, 
b  —  c  =  kxm, 

where  k  and  kx  are  integers,  and  by  addition 

a  —  c=  (k  -\-  kx)m', 

that  is,  a  =  c,  mod  m. 

It  is  now  evident  that  we  can  divide  all  integers  into  classes 
with  respect  to  a  given  modulus,  if  we  put  into  the  same  class 
those  and  only  those  integers  which  are  congruent  to  each  other 
with  respect  to  this  modulus.  We  ask:  How  many  such  classes 
will  there  be  for  any  given  modulus  m? 

Any  integer,  a,  can  be  written  in  the  form 

a  =  km  +  r, 

where  k  is  an  integer  and  r  is  one  of  the  integers 

o,  1,2,3,  •••,|m|— 1. 

But  a  is  congruent  to  r,  mod  m,  and  if  we  give  k  all  integral 
values  from  —  00  to  +00,  the  resulting  values  of  a  will  be  a 
series  of  integers,  all  of  which  are  congruent  to  r,  and  hence  by  i 
to  each  other  with  respect  to  the  modulus  m.  By  putting  for  r 
the  I  m  I  different  values  o,  1, 2,  3,  •••,  |  m  \  —  1,  we  shall  get  |  m  | 
classes  and  every  integer  is  seen  to  fall  into  one  or  the  other  of 
these  classes.  An  integer  can  not  be  in  two  different  classes,  for 
then  we  should  have 

a  =  km  +  r  =  kxm  +  ru 


OF 
i££LlFOBl^ 


THE   RATIONAL   REALM CONGRUENCES.  33 

where  r=¥ri,    - 

which  gives  (  k  —  k1)m  =  r1  —  r. 

Since  the  first  member  of  this  equation  is  divisible  by  m,  the 
second  member  must  be  divisible  by  m  also,  but  since  r  and  rt  are 
both  positive  and  less  than  |  m  |,  we  have  \r  —  rx\  <  \m\,  and 
hence  r  —  rx  is  not  divisible  by  m,  unless  r  —  rx  =  o\  that  is, 
r  —  rx  and  hence  k  =  k1. 

There  are  therefore  exactly  |  m  |  incongruent  number  classes 
with  respect  to  the  modulus  m,  each  integer  being  in  one  and  but 
one  of  the  classes. 

The  absolute  value  of  an  integer,  m,  may  now  be  defined  as  the 
number  of  incongruent  number  classes  with  respect  to  the  mod- 
ulus m. 

This  definition  brings  out  clearly  a  reason  for  the  introduction 
of  the  absolute  value  of  an  integer ;  that  is,  to  express  the  result 
of  an  enumeration  as  a  function  of  an  integer. 

In  all  theorems  relating  to  congruences  we  shall  think  of  the 
entire  system  of  rational  numbers  as  divided  into  such  classes, 
with  respect  to  some  given  modulus ;  and  whatever  is  true  of  any 
individual  integer  with  respect  to  this  modulus  will  be  true  of 
the  entire  class  to  which  it  belongs.  We  shall  thus  deal  rather 
with  the  classes  than  with  the  individuals  in  them  and  it  will  only 
be  necessary  to  have  a  representative  of  each  class. 

Such  a  system  of  |  m  |  representative  integers,  each  integer 
being  chosen  arbitrarily  from  the  class  to  which  it  belongs,  is 
called  a  complete  system  of  incongruent  numbers,  or  a  complete 
residue  system,  with  respect  to  the  modulus  m. 

The  latter  designation  is  derived  from  an  extension  of  the  ordi- 
nary idea  of  the  remainder,  which  holds  when  the  system  chosen 
is  o,  1,2,  ",  \m\ — 1,  by  calling  either  one  of  any  two  integers, 
which  are  congruent  to  each  other  with  respect  to  the  modulus 
m,  a  remainder  or  residue  of  the  other  with  respect  to  m. 

Any  I  m  |  consecutive  integers  evidently  form  a  complete  resi- 
due system  with  respect  to  the  modulus  m. 

The  most  useful  systems  are,  first,  that  composed  of  the  small- 
3 


34  THE   RATIONAL   REALM — CONGRUENCES. 

est  possible  positive  residues 

o,i,2,-",\m\  —  i, 

and  second,  that  composed  of  the  residues  of  smallest  possible 
absolute  value,  the  latter  being,  when m  is  odd  and  |  m  |  =2»  +  i, 

—  n,—  (n  —  i),-..,_  1,0,  i,  •••,«  —  i,«; 

and,  when  m  is  even  and  \m\=2n 

—  (n —  i),  •••, —  1,0, 1,  "'9n — 1,«, 

the  two  residues  n  and  —  n  being  congruent  to  each  other,  mod  m. 
Ex.    If  m  =  ii,  each  of  the  systems 

o,  i,  2,  3,  4,  s,  6,  7,  8,  9,  io; 

—  5,  —4,  —3,  —2,  -M,  o,  i,  2,  3,  4,  s; 
50,  —15,  —25,  20,  32,  22,  —io,  13,  —19,  4,  16 

is  a  complete  residue  system,  mod  II. 

ii.  Addition  and  subtraction  of  congruences. 

If  a1  =  b1,  mod  m,  3) 

and  a2z=zb2,  modm,  4) 

then  a,  ±  o2  aa  frx  ±  b2,  mod  m ; 

for  we  have  from  3)  and  4),  respectively, 

ai  —  &i  =  &iwj 
#2  —  &2  =3  k2mf 

whence  {ax  ±  a2)  —  (&x  ±  &2)  —  (A  ±  &2)w; 

that  is,  (a±  ±a2)^=b1±  b2,  mod m. 

iii.  Multiplication  by  an  integer. 

If  a  =  b,  mod  ni}  5) 

then  ac^bc,  modw; 

for  from  5)  we  have  (a  —  b)=km; 
whence  ac  —  be  =  kem ; 

that  is,  ac^bc,  modw. 

iv.  Multiplication  of  congruences. 

If  ax^bu  modw,  6) 


I 

THE   RATIONAL   REALM CONGRUENCES.  35 

and  a2  =  b2,  modm  7) 

then  a1a2^bjb2i  modm; 

for  from  6)  we  have  by  iii 

axa2  =  bxa2,  modm; 
and  similarly,  from  7)  bra2^bxb2,  modm, 
whence  by  i  axa2  =  bxb2,  mod  m. 

From  this  it  follows,  evidently,  that  if 

a  =  &,  modm, 

then  (&  =  &,  modm, 

where  k  is  any  positive  integer. 

v.  If  f(x)  be  a  polynomial  in  x  with  integral  coefficients; 

that  is,  f(x)=  %*"  +  0i*n_1  +  •  •  •  +  On, 

and  if  r^rx,  modm, 

then  f(r)^=f(rx),  modm,  8) 

for  from  r  =  fi,  modm 

it  follows  by  iv  and  iii  that 

a.rn-i  ==  tfif^*-*,  mod  m,  i  =  o,  1 , 2,  •  •  • ,  w, 

and  by  addition  we  obtain  8). 

It  may  be  shown  similarly  that  if  f(xlf  x2,  '-',xn)  be  a  poly- 
nomial in  xlfx2,  --,Xn  with  rational  integral  coefficients,  and  if 

a2  — ^2    L  modm, 

On  =  in  J 

then  f(a1,a2,'-,an)=f(b1,b2,--,bn),  modm. 

Ex.    Let  f(x)=2x3  —  x*  +  s; 

since  — 3    ebeII,  mod  7, 

we  have  /( — 3)  =/(n),  mod7; 

that  is,  — 58  =  2546,  mod  7. 


36  THE   RATIONAL   REALM — CONGRUENCES. 

vi.  Removal  of  a  common  factor. 

We  have  seen  in  III  that  we  can  multiply  both  members  of  a 
congruence  by  any  integer,  without  affecting  the  validity  of  the 
congruence ;  the  converse  of  this,  however,  is  not  in  general  true. 

Thus  we  have  8  aa  14,  mod  6, 

but  ,  4  4s  7,  mod  6. 

To  consider  this  question  in  general,  let 

a==&,  modw> 

be  a  congruence  in  which  a  and  b  are  both  divisible  by  k ;  that  is, 

a  =  axk  and  b  =  bxk. 

where  ax  and  bx  are  integers. 

Then  from  axk  =  bxk,  mod w, 

it  does  not  necessarily  follow  that 

ax  =  blt  mod  m  ; 

for  that  ax —  bx  shall  be  divisible  by  m  is  not  a  necessary  conse- 
quence of  k(ax  —  bx)  being  divisible  by  m,  unless  k  be  prime  to  m, 
and  all  we  can  say  in  general  is  that  ax  —  bx  is  divisible  by  those 
factors  of  m  which  are  not  contained  in  k ;  that  is,  by  m/d,  where 
d  is  the  greatest  common  divisor  of  k  and  m. 

Hence  from  axk  =  bxk,  mod  m, 

it  follows  in  general  only  that 

ax  m  bv  mod  -^,  9) 

where  d  is  the  greatest  common  divisor  of  k  and  m. 
If  k  be  prime  to  m,  d  is  1,  and  hence  from  9)  we  have 

ax^==bx,  modm. 

Ex.    From  8^14,  mod  6, 

it  follows  that  4^7,  mod  3; 

but  from  5  =  35,  mod  6, 

it  follows  that  1  ==   7,  mod  6. 


THE   RATIONAL   REALM — CONGRUENCES.  37 

vii.  //  a  =  &,  mod m, 

and  d  be  a  divisor  of  m,  then 

a^=b,  mode?; 

for  since  a  —  b  is  divisible  by  m  it  is  divisible  by  d. 
viii.  If  a=b  with  respect  to  each  of  the  moduli  mv  m2,  •••, 

mn,  then  a  =  b,  mod  I, 

where  I  is  the  least  common  multiple  of  mlfm2)  --,mn\  for  since 
a  —  b  is  divisible  by  each  of  the  integers  mx,  m2,  ••,  mn,  it  is  divis- 
ible by  their  least  common  multiple.  An  important  special  case 
of  this  is  when  mltm2,  ••yw*  are  prime  each  to  each,  /  being  then 
their  product. 

ix.  All  integers  belonging  to  the  same  residue  class  have  with 
the  modulus  the  same  greatest  common  divisor;  for  if 

a  =  &,  modw, 

then  a  —  b  =  km; 

and  any  integer  that  divides  a  and  m  must  also  divide  b,  and  any 
integer  that  divides  b  and  m  must  also  divide  a.  Therefore  the 
greatest  common  divisor  of  a  and  m  is  identical  with  the  greatest 
common  divisor  of  b  and  m.  In  particular  //  any  integer  of  a 
residue  class  be  prime  to  the  modulus  m,  then  all  the  integers  of 
this  class  are  prime  to  m. 

§2.    The  Function  <f>(m). 

By  ^(m)1  we  denote  the  number  of  integers  of  a  complete 

residue  system,  mod  m,  which  are  prime  to  m.     Such  a  system 

of  integers  is  called  a  reduced  residue  system,  or  a  reduced  system 

of  incongruent  numbers,  mod  m.     That  the  number  of  integers 

in  such  a  system  is  independent  of  the  complete  residue  system 

chosen  is  obvious  from  §  i,  ix.     We  can  therefore  calculate  <£(w) 

for  a  particular  value  of  m  by  writing  down  any  complete  residue 

system,  mod  m,  and  removing  those  integers  of  this  system  that 

are  not  prime  to  m.     The  number  of  those  remaining  is  evidently 

4>(m). 

1  The  symbol  is  due  to  Gauss :  Disq.  Arith.,  §  38,  Works,  Vol.  I.     Euler 
first  gave  a  general  expression  for  <t>{m)  :  Comm.  Arith.,  I,  p.  274. 


38  THE   RATIONAL   REALM CONGRUENCES. 

Thus  for  m  =  —  10,  take  as  a  complete  residue  system 

—  10,  —  19, 2,  —  7,  —  16,  5, 16, 17, 18,  —  1. 

Striking  out  the  integers  — 10,2,  — 16,5,16,18,  that  are  not 
prime  to  —  10,  we  have  left  the  four  integers  —  19, —  7, 17, —  1, 
that  constitute  a  reduced  residue  system,  mod  — 10. 

Hence  <j>( —  10)  =  4. 

As  a  second  example,  let  m  =  7. 

A  complete  residue  system,  mod  7,  is 

0,1,2,3,4,5,6, 

and  we  see  that  <f>(y)=6. 

The  last  example  leads  Us  at  once  to  a  general  expression  for 
<t>(p),  when  p  is  a  prime  ;  for  the  integers  o,  I,  •••,  |  p  |  —  1  con- 
stitute a  complete  residue  system,  mod  p,  and  are,  with  the  excep- 
tion of  o,  all  prime  to  p,  whence  it  is  evident  that 

<t>(P)=-\P\  —  1. 

It  should  be  observed  that,  since  the  units  are  regarded  as 
prime  to  themselves, 

4>(±i)=i. 

The  first  method,  which  we  shall  employ  to  obtain  a  general 
expression  for  <f>(m)  in  terms  of  m,  is  exactly  similar  to  that  em- 
ployed in  the  examples  just  given ;  that  is,  we  write  down  a  com- 
plete residue  system,  mod  m,  remove  those  integers  of  this  system 
which  have  a  common  divisor  with  m,  and  count  those  remaining, 
their  number  being  <f>(m). 

The  general  expression  for  <f>(m),  where  m  is  any  integer,  is 
given  by  the  following  theorem : 

Theorem  i.  //  plt  p2>  '-,pr  be  the  different  positive  prime  fac- 
tors of  m,  and  <f>(m)  denote  the  number  of  integers  of  a  complete 
residue  system,  mod  m,  that  are  prime  to  m,  then 

^(m)  =  |m|(i-f)(i-^)-(i-^-). 

Pi  P2  Pr 

Since,  evidently, 

<f>(—m)=<f>(m), 

we  can  without  loss  of  generality  assume  m  positive. 


THE   RATIONAL   REALM — CONGRUENCES.  39 

Let 

tn  =  p1eip2e2'-  prer, 

where  px,p2,  '",pr  are  the  different  positive  prime  factors  of  m. 
Take  as  a  complete  residue  system,  mod  m, 

i,2,3,4,---,m  S) 

Our  task  is  to  remove  from  the  system  S  those  integers  which 

are  divisible  by  one  or  more  of  the  primes  pltp2,  '",pr,  and  to 

count  the  integers  left.     We  shall  first  remove  those  divisible  by 

plt  namely  the  m/p1  integers 

m 
Pi,2px,2>Px>'->irPx> 

Vx 
Removing  these  from  S  there  remains  a  system,  Slf  consisting 
of  m  —  tn/plt  —rn(i  —  l/P%),  integers,  none  of  which  is  divis- 
ible by  px. 

From  this  system  Sx  we  must  now  remove  those  integers  that 
are  divisible  by  p2 ;  that  is,  those  integers  of  5  which  are  not  divis- 
ible by  px  but  are  divisible  by  p2.  The  integers  of  S  which  are 
divisible  by  p2  are  the  m/p2  integers 

m 
p2>2p2,3p2,'--,rp2,---,  —  p2, 
A 

and  the  necessary  and  sufficient  condition  that  any  one,  rp2,  of 

these  integers  be  also  divisible  by  plt  is  that  the  coefficient,  r,  of 

p2  shall  be  divisible  by  fv 

The  number  of  the  integers,  which  are  to  be  removed  from  the 

system  S\  on  account  of  their  divisibility  by  p2,  is  therefore  the 

same  as  the  number  of  the  integers 

m 
1,2,3,   ~,  ~> 
A 

which  are  not  divisible  by  plt  and  this  is,  since  m/p2  is  divisible 

by  px,  exactly  as  in  the  first  step  of  this  proof 


j\ 


\  m 


t-$ 


40  THE   RATIONAL   REALM — CONGRUENCES. 

There  remains  then  of  S  a  system,  S2,  of 


integers,  none  of  which  is  divisible  by  pr  or  p2.  We  are  now  led 
to  conclude  by  induction  that  the  number  of  the  integers  of  S, 
which  are  divisible  by  none  of  the  r  primes  plt  p2,  •-,  pr  is 


m 


('4)(-i)-(-7.) 


m 


To  prove  that  this  is  correct,  it  is  only  necessary,  since  we  know 
that  it  holds  for  r  =  2,  to  show  that,  if  it  holds  for  r  =  i,  it  holds 
for  r  =  i-\-  i. 

Assume  then  that,  having  removed  from  5*  the  integers  divisible 
by  one  or  more  of  the  i  primes  plt  p2,  '",pi,  there  is  left  a  system 

(-*)(-*)•  •■(-*)      | 

integers. 

To  obtain  the  number  of  integers  of  S  that  are  divisible  by 
none  of  the  primes  p1}p2,  '",pi+i,  we  must  remove  from  Si  those 
integers  which  are  divisible  by  pi+1  and  count  those  remaining. 
The  integers  of  Si  that  are  divisible  by  pi+1  are  the  same  as  the 
integers  of  5  that  are  divisible  by  pt+1  but  are  divisible  by  none 
of  the  primes  p1}p2,  •••,  pi.  The  integers  of  S  that  are  divisible 
by  pi+1  are 

m 

Pi+v  2A+i,  • ' '»  rA+v  ' '  ••  J-Pi+v 

Pi+\ 

and  the  necessary  and  sufficient  condition  that  any  one  rpi+1  of 
these  integers  shall  be  divisible  by  none  of  the  primes  plfp2t  '",pi 
is  that  the  coefficient,  r,  of  pi+1  be  divisible  by  none  of  these  primes. 
The  number  of  integers  to  be  removed  from  St  coincides  there- 
fore with  the  number  of  the  integers 

m 

A+i 


THE   RATIONAL   REALM CONGRUENCES.  4 1 

that  are  divisible  by  none  of  the  primes  plf-",pi'     By  formula 
i),  whose  correctness  has  been  assumed,  this  number  is 


t(l-.k)(l-d-(l-lp} 


m 

A 

Subtracting  this  number  from  i)  we  get 

■■i'-k)-(-j)(-k){-k)(-^ 

an  expression  identical  in  form  with  i),  as  the  number  of  the 
integers  oi  5*  which  are  divisible  by  none  of  the  primes 

Pi,p2>'--,pi,pi+i- 

But  we  have  proved  the  correctness  of  I )  when  i  =  2,  hence  the 
theorem  holds  when  t=3,  and  similarly  when  i  =  r. 

If  m  be  any  integer,  positive  or  negative,  and  plt p2,  •■■Jpr  be 
its  different  prime  factors,  positive  or  negative,  we  have  as  an 
absolutely  general  expression  for  <f>(m) 

^)  =  IH(I-f^)-(I-^r). 

Making  use  of  the  representation  of  m  as  a  product  of  powers  of 
its  different  prime  factors,  we  obtain  another  expression  for 
<f>(m)  ;  that  is, 

<t>(m)  =  (\pi\--l)\pi\e^  --  (\pr\  —  l)\pr\er-K 

If  m  be  a  power  of  a  single  prime  as  pe,  we  have 

<t>(±p')  =  (\p\-i)\p\e-\ 
and,  in  particular,  when  e=i, 

<t>(P)  =  \p\  —  i- 
Ex.    Let  m  =  60  =  22  •  3  •  5. 

We  have  0(60)  =  60(1  —  i)  (1  —  *)  (1  —  |) 


42  THE   RATIONAL   REALM CONGRUENCES. 

a  result  seen  to  be  true  when  we  write  down  the  complete  residue  system, 

mod  60,  1,  2,  •  •  •,  60. 

For  when  we  remove  those  integers  which  are  not  prime  to  60,  there 
are  left  the  integers 

1,  7,   11,   13,   17,   19,  23,  29,  31,  37,  41,  43,  47,  49,  53,  59, 

in  number  16. 

We  observe  that  <f>(m)  is  an  even  number  except  when  m  =  ±  1, 
or  ±2;  for  if  m=±2e,  we  have  <f>(±  2e)  =2e~1,  which  is  an 
even  number  when  e  >  1,  and  if  m  contain  an  odd  prime  factor 
plt  then  from  2)  it  is  evident  that  <f>(m)  contains  the  even  number 
I  p  I  —  1  as  a  factor  and  hence  is  an  even  number.  This  may  be 
proved  independently  of  the  formula.1 

The  above  proof,  which  is  the  one  usually  given  for  this 
theorem,  has  been  used  here  on  account  of  its  great  simplicity. 
It  does  not,  however,  admit  of  extension  to  the  higher  realms  in 
Jhe  form  here  given,  since  a  property  of  rational  numbers  has 
been  made  use  of  which  has  no  analogue  in  the  case  of  algebraic 
numbers  of  a  higher  degree.  We  therefore  give  below  a  proof 
depending  upon  the  same  principles  as  the  above  but  so  formu- 
lated that  it  is  at  once  capable  of  extension  to  a  realm  of  any 
degree.2  In  giving  these  two  forms  we  hope  to  make  clear  to  the 
reader  some  of  those  conditions  which  must  be  satisfied  by  the 
form  of  proof  of  a  theorem  regarding  rational  integers  in  order 
that,  should  the  theorem  be  found  to  hold  for  the  integers  of  any 
algebraic  number  realm,  the  same  form  of  proof  can  be  used 
for  it  in  the  general  case.  The  proof  of  the  general  theorem 
(Th.  1)  depends  directly  upon  the  following  simple  theorem: 

Theorem  2.  //  a=bc,  where  b  and  c  are  any  integers,  there 
are  in  a  complete  residue  system,  mod  a,  exactly  \c\,  =  |a/&|, 
numbers  that  are  divisible  by  b. 

Since  by  §1,  ix,  if  the  theorem  be  true  for  any  particular 
residue  system,  mod  a,  it  is  true  for  all,  we  shall  construct  |c| 
numbers  which  are  divisible  by  b  and  incongruent  each  to  each, 
mod  a,  and  shall  then  show  that  no  other  number  of  a  complete 

1  Cahen :  p.  33.  2  See  p.  44. 


THE   RATIONAL   REALM — CONGRUENCES.  43 

residue  system,  mod  a,  of  which  these  numbers  are  a  portion,  can 
be  divisible  by  b. 

Let  clfc2,"-,cCf  2) 

be  any  complete  residue  system,  mod  c.     The  integers 

bclfbc2,--,bcc  3) 

are  incongruent,  mod  a,  for  if 

ben  ass  bci,  mod  a, 
then  ch  =  Ci,  mode, 

which  is  impossible. 

Moreover,  every  integer,  bd,  divisible  by  b  is  congruent,  mod  a, 
to  some  one  of  the  numbers  3),  for  d  is  congruent,  mod  c,  to  some 
one,  say  Ci,  of  the  integers  2),  and  from 

d  =  Ci,  mode, 

it  follows  that  bd  =  bci,  mod  a,  and  bd  is  one  of  the  integers  3)! 
Hence  the  integers  3)  comprise  all  those  integers  of  a  complete 
residue  system,  mod  a,  of  which  they  are  a  portion,  that  are  divis- 
ible by  b.  They  are  |  c  |  in  number  and  the  theorem  is  therefore 
proved. 

If  we  select  the  particular  residue  system 
1,  2,  •••,  \m\, 
and  observe  that  the  integers  of  this  system,  that  are  divisible  by  b,  are, 


considering  b  positive,  b,  2b, 


b, 


the  truth  of  the  theorem  is  at  once  evident.  The  form  of  proof  used 
above  has,  however,  been  chosen  on  account  of  its  immediate  adaptability 
to  the  higher  realms. 

From  the  above  theorem  we  obtain  at  once  the  following : 


Theorem  3.     If  p  be  any  prime 


There  are  in  a  complete  residue  system,  mod  pe,  exactly  |  pe/p  | 
numbers  that  are  divisible  by  p  and  therefore  \  pe  \  —  |  pe/p  \  that 
are  prime  to  p.     Hence  the  theorem. 


44  THE   RATIONAL   REALM CONGRUENCES. 

We  shall  now  prove  again  Theorem  i,  placing  no  restriction 
upon  either  m  or  its  prime  factors  as  to  sign. 

Theorem  i.  If  pu  p2,  •  •  •,  pr  be  the  different  prime  factors  of 
m,  and  <j>(m)  denote  the  number  of  integers  of  a  complete  residue 
system,  mod  m,  that  are  prime  to  m,  then 

Second  Proof.1 

Denote  by  S  a  complete  residue  system,  mod  m,  and  let 

\m\ . _.  | m ; J  \m\ 

\m\  \m\  \m\ 

S.  =  rrrrr,  +  rrrrn  +  •  ■  ■  + 


AllAl     lAllAl  lA-.llAl' 


5  = 


AllAh--|Al' 

Consider  now  the  sum 

N=\m\-S1JrS:!---  +  (-iySr. 

Making  use  of  Theorem  2,  we  see  that  an  integer  of  S,  which  is 
divisible  by  i  of  the  />'s  but  not  by  *  +  1  of  them,  is  counted 
once  in  |  m  |,  %  times  in  Slf  i(i —  i)/i-2  in  S2,  -•-,  and  finally  once 
in  Si.     Hence  this  integer  contributes  to  N  the  number 

i-i+i±i^--  +  (-iy=(i-iy=o. 

Therefore  every  integer  of  S  that  is  not  prime  to  m  contributes  o 
to  N,  while  every  integer  oi  S  that  is  prime  to  m  contributes  1  to 
N,  since  it  is  counted  once  in  \m\  and  is  not  counted  in  Sly  S2,  •••, 
Sr.  Hence  N  is  the  number  of  those  integers  of  5*  which  are 
prime  to  m;  that  is, 

N  =  <f>(m). 
1  Mathews:  §7. 


THE   RATIONAL   REALM — CONGRUENCES.  45 

Therefore 

+(«)  =  |m|—  St  +  S, +  (-iysr 


=]Ml(l-\k\)(l-\k\)-{l-\k\} 


§  3.    The  Product  Theorem  for  the  ^-Function. 

Theorem  4.  //  m  =  w1w2,  where  mx  and  m2  are  prime  to 
each  other,  then  <f>(m)  =  cf>(m1)cf>(m2). 

Let  m1  =  ±  p1eip2e2 •  • '  prer, 

and  m2  =  ±  qxflqj* ' '  ■  fA 

where  plf  p2,-',  pr,  qlf  q2r",q8  are  different  primes. 

Then  m  =  ±  pt*  •  •  •  prer  qxfl  ■  ■  ■  #/', 

and 

^«)  =  l,«l(I-fii])...(I-^|)(.-|^)...(I-^) 
=w(i-^])-(-^)i-2i(i-^i)-(-^) 

Ex.    Since  60  =4-15,  and  4  is  prime  to  15,  we  have 
0(6o)  =0(4)0(15)  =2-8=16 

The  above  result  can  evidently  be  extended  to  a  product  of 
any  number  of  factors,  which  are  prime  each  to  each;  that  is,  if 
m  =  m1  m2  ■  •  •  mr,  where  mlt  m2,  •  •  •  mr  are  prime  each  to  each, 

then  <f>(m)  =<f>(m1)(f>(m2)  •••  <f>(mr). 

This  theorem  is  useful  in  the  calculation  of  <f>(m). 

Ex.     Since  315  =  32  •  5  •  7,  we  have 

0(315)  =  0(32)0(5)0(7)  =  6-4-6  =  144. 

This  property  of  the  function  <f>(m)  can  be  derived  without  the 
use  of  Theorem  1.     This  having  been  done  and  having  shown  that 


♦eo-i'iO-lTl)' 


46  THE   RATIONAL   REALM CONGRUENCES. 

we  can  derive  the  general  expression  for  <f>(m)  in  terms  of  m. 
This  is  the  method  adopted  by  Gauss.1 

§  4.     The  Summation  Theorem  for  the  ^-Function. 

Theorem  5.  //  d  be  any  divisor  of  m  and  m  =  nd,  the  num- 
ber of  integers  of  a  complete  residue  system,  mod  m,  which  have 
with  m  the  greatest  common  divisor  d  is  <f>(n). 

Since  by  §  1,  ix,  if  the  theorem  be  true  for  any  particular  resi- 
due system,  mod  m,  it  is  true  for  all,  we  may  take  the  system  used 
in  Theorem  2.     We  have  shown  there  that  the  system  of  integers 

dn^dn^-'-ydnn,  1) 

where  nlfn2i-  -,nn  is  a  complete  residue  system,  mod  n,  com- 
prises all  those  and  only  those  integers  of  a  complete  residue  sys- 
tem, mod  m,  which  are  divisible  by  d. 

Hence  the  integers  of  this  complete  residue  system,  mod  m, 
which  have  with '^  the  greatest  common  divisor  d  are  those  of  the 
system  1)  in  which  the  coefficient  of  d  is  prime  to  n.  Since 
nlfn2,  -",nn  is  a  complete  residue  system,  mod  n,  the  number  of 
these  integers  is  <}>(n)  and  the  theorem  is  proved. 

Theorem  6.  If  d17d2f-",  dr  be  the  different  divisors  of  m,  we 
have 

Y>(«*«)  =  M. 

The  proof  of  this  theorem  follows  easily  from  the  last.  Write 
down  all  the  different  divisors,^,  d2,  •••,  dr,  of  the  integer  m, 
and  let 

m  =  mxdx  =  m2d2  ==•••  =  mrdr, 

observing  that  both  1  and  m  are  included  among  the  divisors  of 
m.  Separate  the  integers  of  a  complete  residue  system,  mod  m, 
into  classes  in  the  following  manner.  Place  in  the  first  class  those 
integers  of  the  system  that  have  with  m  the  greatest  common 
divisor  dx ;  by  Theorem  5  they  will  be  ^(wj  in  number.  Place 
in  the  second  class  those  integers  of  the  system  that  have  with 
m  the  greatest  common  divisor  d2 ;  they  will  be  similarly  <f>(m2) 

1  Disq.  Arith.,  Art.  38.     Works,  Vol.  I.     See  also  p.  75. 


THE   RATIONAL   REALM CONGRUENCES.  47 

in  number.  Proceeding  in  this  way  it  is  evident  that  we  shall 
have  r  classes  and  that  each  integer  of  the  system  will  occur  in 
one  and  but  one  of  these  classes.  But  the  number  of  integers  in 
a  complete  residue  system,  mod  m,  is  \m\.  Hence  the  total 
number  of  integers  in  these  classes  is  \m\.  Since,  however,  the 
total  number  of  integers  in  the  classes  is  also 

^(mj  +  <f>(m2)  +  •••  +<£(mr), 

and  mXimv'",mr 

are  merely  dlf  d2,'--,dr 

in  different  order,  we  have 

f>(*)'sH«l- 

i=l 

Ex.    Let  m  =  30.     The  different  divisors  of  m  are 

1,  2,  3,  5,  6,  10,  15,  30. 
We  have  then 

0(1)  +0(2)  +  0(3)  +  0(5)  +  *(6)  +  0(io)  +  0(15)  +0(30)  =  30, 

a  result  which  may  be  verified  by  calculating  the  values  of  0(i),  0(2), 
•  •  •>  0(30) »  and  taking  their  sum.     We  have 

1  +  1+2  +  4  +  2  +  4+8  +  8  =  30. 

The  above  property  of  the  function  <f>(m)  has  been  derived 
immediately  from  the  original  definition  of  the  function,  no  use 
having  been  made  of  the  expression  found  for  <f>(m)  in  terms  of 
m.  It  completely  defines  <f>(m)  and  from  it  we  can  derive  all  the 
properties  of  the  function,  in  particular  the  expression  for  <f>(m) 
in  terms  of  m.1 

We  shall  give  now  another  proof  of  this  property  of  <f>(nt) 
making  use  of  Theorems  3  and  4. 

In  order  to  bring  out  clearly  the  analogy  which  exists  between 
this  proof  and  that  of  the  corresponding  theorem  in  the  higher 
realms  which  will  be  given  later  we  shall  put  no  restriction  upon 
either  m  or  its  prime  factors  as  to  their  sign,  although  so  far  as 
this  proof  is  concerned  merely  with  rational  integers,  they  may 
evidently  all  be  assumed  positive  without  limiting  its  generality. 

1  Dirichlet-Dedekind :  §  138. 


48  THE  RATIONAL   REALM CONGRUENCES. 

Let  m=±  pxeip2e*  • ' '  prer 

where  p!,p2,"',pr  are  different  primes. 
Every  divisor  of  m  has  the  form 

di  =  ±px^p^--'prfr  i) 

where  fx  is  one  of  the  numbers  o,  I,  •••  *£, 

f2  is  one  of  the  numbers  o,  i,  •••  e2, 


fr  is  one  of  the  numbers  o,  i,  •••  er. 
We  have  by  Theorem  4 

♦  (A) =+(P%H)4>{PtU)  '"4>{Prfr)-  2) 

If  we  let  flf  f2,"-,fr  run  through  the  values  o,  1,  •  •  •,  e1 ;  o,  1,  •  •  •,  e2 ; 
•••  ;o,  1,  --',er,  respectively,  we  obtain  from  1)  all  the  divisors  of 
m,  and  from  2)  the  corresponding  values  of  <f>(di)  whose  sum  is 

1=1 

We  see  therefore  that  the  terms  of  the  series  obtained  by  multi- 
plying out  the  product 

P=  [#(x)  +  *(*)  +  *<*■)  +  •••  +  4>(/>1e0]  - 
are  identical  with  the  terms  of 

I>(rfO ; 

»=l 
r 

that  is,  P  =£<£(</,). 

i=l 

But 

*(i)  =  i,  <t>(p1)  =  \p1\-h'-,  <f>(Piei)  =  \Pi\e>-1(\Pi\  —  i), 
whence 

*(0  +  <£(/>i)  +  •••  +  4>(/>ie0  =  I  />i  |eS 
and  similarly  for  the  other  factors  of  3). 

Therefore 

P=\Pi\e*\P1\e*--\pr\e'=\m\, 


THE   RATIONAL   REALM CONGRUENCES.  49 

and  hence 

±<t>(di)  =  \m\. 

§  5.  Discussion  of  Certain  Functional  Equations  and  Another 
Derivation  of  the  General  Expression  for  <j>(m). 

Theorem  7.  //  m  be  any  integer  other  than  ±  1,  whose  dif- 
ferent prime  factors  are  p.lt  p2,  '">  pr,  and  d  any  divisor  of  m  other 
than  dz  m,  and  if  we  separate  all  integers  of  the  form 

m 

Pl,P2  '"Pi 

no  p  being  repeated,  into  two  classes,  I  and  II,  putting  in  class  I 
those  such  that  m  is  divided  by  none  or  by  the  product  of  an  even 
number  of  the  p's,  and  in  class  II  those  such  that  m  is  divided  by 
the  product  of  an  odd  number  of  the  p's,  then  exactly  as  many 
integers  of  the  one  class  are  divisible  by  d  as  of  the  other.1 

Before  proving  this  theorem  it  will  be  well  to  illustrate  its 
content  by  an  example. 

Let 

m  =  6o  =  22.3.5. 

Forming  the  above  mentioned  numbers  we  have  the  following : 

^         _    .      60    60    60     ,       .      _ 
Class    1 :  00,  — ,  — ,  —  ;  that  is,  00, 10, 6, 4. 
2.3    2-5    3-5 

_,       TT     60  60  60     60       ,      . 

Class  II:   — ,  — ,  — , ;  that  is,  30,20, 12,2. 

2      3      5     2.3.5'  '° 

If  now  d=io,  we  see  that  two  numbers  of  each  class  are 
divisible  by  10;  that  is,  60  and  10  of  I,  and  30  and  20  of  II. 

We  proceed  to  prove  the  theorem,  observing  that  since  we  are 
concerned  here  only  with  questions  of  divisibility  and  since  in 
such  questions  what  is  true  of  one  associate  of  an  integer  is  true 
of  both  of  its  associates,  we  may  without  limiting  the  generality 
of  our  proof  assume  m,px,  --ypr  and  d  to  be  positive. 

Making  this  assumption,  we  see  that  the  positive  and  negative 
terms  of  the  developed  product 

1  Dirichlet-Dedekind :  §  138. 
4 


50  THE   RATIONAL   REALM CONGRUENCES. 


VI 


(<-&(->)■;•('-*)        ■> 

coincide  respectively  with  the  integers  of  I  and  II.  That  is, 
denoting  by  %mlt  2m2,  respectively,  the  sums  of  the  numbers  of 
these  classes,  we  have 

Let 

we  shall  first  prove  the  theorem  for  the  case  in  which 

C1  =  €2  =  ' ' '  ==  €r  =  I  J 

that  is,  m  is  not  divisible  by  a  higher  power  than  the  first  of  any 
prime. 

Setting  pxp2  •  •  •  pr  =  a,  we  have 

a(I-71)(I-72)-(I-i)=(A-I)^-I)-^-I) 

tm  2^  —  2tf2> 

where  ^ax,^a2  have  meanings  corresponding  to  those  of  ^mXi^m2. 
If  now  b  be  any  positive  divisor  of  a  other  than  a,  the  number 
of  the  ax  terms  that  are  divisible  by  b  is  exactly  equal  to  the  num- 
ber of  a2  terms  that  are  divisible  by  b,  for,  if  we  put 

a  =  bq1q2  •••  q8 

where  qlt q2,'">q8  are  those  prime  factors  of  a  which  do  not 
divide  b,  then  the  ax  terms  and  the  a2  terms  that  are  divisible  by  b 
are  respectively  the  positive  and  negative  terms  of  the  developed 
product 

b(q1  —  i)(q2—i)  ••'  (?•—  I).  2) 

Moreover,  since  b=%=a  there  is  at  least  one  prime,  q,  that  di- 
vides a  but  not  b ;  that  is,  there  is  at  least  one  q.  Hence  there 
are  exactly  as  many  positive  as  negative  terms  in  the  developed 
product  2)  and  consequently  as  many  of  the  a/s  as  of  the  a2,s 
are  divisible  by  b. 


THE    RATIONAL   REALM CONGRUENCES.  5  I 

The  theorem  is  therefore  proved  for  the  case  in  which  m  is  not 
divisible  by  a  higher  power  than  the  first  of  any  prime. 

We  proceed  now  to  prove  the  theorem  for  the  general  case. 
Let  a,  alf  a2  retain  the  meanings  assigned  above.     We  have 

m  =  p^p^1  •  •  ■  prer-1p1p2  •  •  •  pr  =  na, 

and  it  is  evident  that  the  integers  mlf  m2  coincide  respectively 
with  the  products  nalt  na2.  Now  let  d  be  any  positive  divisor  of 
m  other  than  m  and  let  g  be  the  greatest  common  divisor  of  the 
two  integers 

d  =  gb,  n  =  gc. 


We  see  that  b  is  a  divisor  of  a ;  for  ca/b  is  an  integer  since 

3) 


ca      gca      na      m 


which  is  an  integer,  and  c  is  prime  to  b. 

From  3)  it  follows,  since  c  is  prime  to  b,  that,  if  d  =  m,  then 
c=i  and  b  =  a.  Conversely,  if  b  be  equal  to  a,  and  hence  be 
divisible  by  all  prime  factors  of  m,  then  c  must  be  I,  since  it  is  a 
divisor  of  m  but  prime  to  b,  and  hence  d  =  tfk 

Excluding,  therefore,  the  case  d  =  m,  so  that  we  have  always 
b=$=a,  there  are  among  the  integers  Oj  exactly  as  many  that  are 
divisible  by  b  as  there  are  among  the  integers  a2. 

Since,  moreover,  the  necessary  and  sufficient  condition  that  an 
integer  mls  or  m2,  where 

m1  =  na1  =  gca1, 

or  m2  =  na2  =  gca2 , 

shall  be  divisible  by  d  =  gb,  is  that  a1?  or  a2,  shall  be  divisible  by 
b,  there  are  exactly  as  many  of  the  integers  mx  divisible  by  d  as 
of  the  integers  m2. 

The  theorem  is  therefore  proved. 

Many  interesting  applications  may  be  made  of  this  theorem; 
among  them  are  the  two  following : 


52  THE   RATIONAL   REALM CONGRUENCES. 

Theorem1  8.  A)  If  f(m)  and  F(m)  be  two  functions  of  an 
integer  m  that  are  connected  by  the  relation 

Sf(d)=F(m),  4) 

where  d  runs  through  all  divisors  of  m  including  m,  then 

f(m)  =  ^F(m^)  —  SF(w2),  5) 

where  mlf  m2>  run  through  the  values  defined  in  the  last  theorem. 
B)  If  f(m)  and  F{m)  be  connected  by  the  relation 

Uf(d)==F(m)  6) 

where  the  product  relates  to  the  values  of  the  function  corre- 
sponding to  all  the  values  of  d,  then 

'(jw)=nFky  7) 

To  prove  A  it  is  sufficient  to  observe  that  if  d  be  any  divisor 
of  m  other  than  ±  m,  it  is  a  divisor  of  exactly  as  many  of  the 
■m/s  as  of  the  w2's  (Theorem  7),  and  hence,  when  in  5)  we 
replace  the  F's  by  their  values  in  terms  of  the  /'s  from  4),  f(d) 
will  occur  exactly  as  often  with  the  plus  sign  as  with  the  minus 
sign. 

Hence  all  terms  in  the  second  member  of  5)  will  cancel  except 
f(m)  which  occurs  once  only.  We  shall  illustrate  this  by  a 
numerical  example. 

Letw=i5.    We  have 

15(1  -i)  (1  -i)  =  1-3-5  +  15  =  1  +  15-  (3  +  5), 
whence  ^mx  =1  +  15, 

and  2m2  =  3  +  5. 

Also  from  4) 

/(i)+/(3)+/(5)+/(i5)=f(i5). 
/(i)+/(S)  =-P(5), 

/(0+/(3)  =F(3), 

/(i)  =F(i). 

1  This  theorem  holds  also  in  the  case  m  =  1,  which  was  excluded  in  Th. 
7,  if  we  assume  that  in  this  case  there  is  only  a  single  mh  =  1,  and  no  Iff* 


THE   RATIONAL   REALM — CONGRUENCES.  53 

We  have  now  from  5) 

/(i5)=2F(7»1)-SF(W,2); 
for 

/(i5)=F(i)  +F(i5)-  [F(3)  +F(5)] 

=  /(i)+/(0  +/(3)  +/(5)  +/(I5) 

-(/(i)+/(3)+/(i)+/(5)) 

=  /(iS)- 

The  proof  of  B  is  evidently  exactly  like  that  of  A.     It  will 
suffice  if  we  illustrate  it  by  a  numerical  example. 
Let  w==  15  ;  we  have  from  6) 

/(i)/(3)/(S)/(i5)=^(i5), 
/(i)/(5)  =F(5), 

/(i)/(3)  =-F(3), 

/(i)  =F(i). 

From  7) 

_F(i)F(iS) 
_^(3)>"(5)"' 

_/(i)-/(i)/(3)/(5)/(i5) 
/(iV(3)-/(0/(S)     ' 

—/(M). 

From  Theorem   8,  A,  we  can   easily   deduce  by  the  aid  of 
Theorem  6  the  general  expression  for  <f>(m). 
From  Theorem  6  we  have 

where  d  runs  through  all  divisors  of  m. 
Applying  Theorem  8,  we  have 

f(m)  =  <£(w)  and  F(m)  =  |  m  |, 


54  THE   RATIONAL    REALM CONGRUENCES. 

and  hence 

W=^I-2^  =  |OT|(I-^)(I-^-|)...(I-r/i-|). 

As  an  example  of  the  use  of  Theorem  8,  B,  we  give  the  fol- 
lowing : 

Let  f{m)  =  p,  when  m  is  a  power  of  the  prime  number  p,  and 
f(m)  =  i,  when  m=i  or  is  divisible  by  two  or  more  different 
prime  numbers. 

We  have 

n/(<o=m, 

where  d  runs  through  all  divisors  of  m,  from  which  it  follows  by 
Theorem  8,  B,  that  the  quotient 

— —  =  /(m) 

is  different  from  I  only  when  m  is  a  power  of  a  prime  number, 
in  which  case  it  is  equal  to  this  prime. 

For  a  derivation  by  another  method  of  the  other  properties  of 
the  <f>  functions  from  the  single  one  that 

%+(d)  =  \nt\t 

see  Kronecker,  Vorlesungen  uber  Zahlentheorie,  Vol.  I,  pp.  245, 

246. 

Also  for  another  independent  proof  that 

+(ofr)»+(a)+(*), 

if  a  be  prime  to  b,  see  the  same,  p.  125. 

§  6.     ^-Functions  of  Higher  Order.1 

The  theory  of  the  <£- function  may  be  generalized  as  follows : 
By  <f>n(m)  we  denote  the  number  of  sets  of  n  integers  of  a  com- 
plete residue  system,   mod  m>,  such   that  the  greatest  common 
divisor  of  the  integers  of  each  set  is  prime  to  m,  two  sets  being 
different  if  the  order  of  the  integers  in  them  be  different. 
For  example,  let  w  =  6;  then 

1,2,3,4,5,6  1) 

1Cahen:    pp.  36,  Z7-     Bachman :    Niedere  Zahlentheorie,  pp.  91,  93. 


THE   RATIONAL   REALM — CONGRUENCES.  55 

will  be  a  complete  residue  system,  mod  6.     All  possible  sets  of 
two  numbers  each  that  can  be  formed  from  the  numbers  i)  are 


I,  I 

h  2 

h  3 

I,  4 

i,  5 

i,6 

2,    I 

2,   2 

2,   3 

2,  4 

2,  5 

2,  6 

3,  i 

3,  2 

3,  3 

3,  4 

3,  5 

3,6 

4,  i 

4,   2 

4,  3 

4,  4 

4,  5 

4,6 

5,  i 

5,  2 

5,  3 

5,  4 

5,  5 

5,6 

6,  i 

6,2 

6,  3 

6,4 

6,  5 

6,  6 

Of  these  there  are  twelve  sets  the  greatest  common  divisor  of 
the  numbers  of  each  of  which  is  not  prime  to  6 ;  they  are 

2,  2 ;  2,  4 ;  2,  6 ;  3,  3 ;  3,  6 ;  4,  2 ;  4,  4 ;  4,  6 ;  6,  2 ;  6,  3 ;  6,  4 ;  6,  6. 

There  are   therefore   twenty- four   sets,   the   greatest  common 
divisor  of  the  numbers  of  each  of  which  is  prime  to  6.     Hence 

4>2(6)=24. 
It  can  be  shown  that 

«^-l*K^Kf)('-Rp)-"(iT.Kp)( 

where  plf  p2,  '",pr  are  the  different  prime  factors  of  m. 
The  following  theorems  can  also  be  proved : 
i.  If  m^p,  a  prime  number,  then 

<f>n(P)  =  \p\n—^ 

ii.  //  \m\  >  2,  <f>n(m)  is  even. 

iii.  If  mx  and  m2  be  two  integers  prime  to  each  other,  then 

4>n(mxm2)  =  <£n(wi)<Mw*)- 
iv.  If  d  run  through  all  divisors  of  m, 
$<j>n(d)  =  \m\n. 

Ex.    Let  m  =  6,  and  n  =  2 ;  then 

02(6)=62(i-J2)(i-J2)=24. 


$&  THE   RATIONAL   REALM CONGRUENCES. 

§  7.  Residue  Systems  Formed  by  Multiplying  the  Numbers 
of  a  Given  System  by  an  Integer  Prime  to  the  Modulus. 

Theorem  9.  //  mlym2f  --,mm  be  a  complete  residue  system, 
mod  m,  and  a  be  prime  to  m,  then  am^  am2,  •••,  amm  is  also  a  com- 
plete residue  system,  mod  m. 

The  integers  amx,  am2,  •  •  • ,  amm  are  incongruent  each  to  each, 
mod  m,  for  from 

ami  =  amj,  modm, 

it  would  follow  that,  since  a  is  prime  to  m, 

miz==mj,  mod  w, 

which  is  contrary  to  the  hypothesis  that  mx,m2,--,mm  form  a 
complete  residue  system,  mod  m.  The  integers  amx,  •  •  •,  amm  are, 
moreover,  |  m'\  in  number.  They  form,  therefore,  a  complete 
residue  system,  mod  m. 

Cor.  If  r1,r2,---,r4>(m)  form  a  reduced  residue  system,  mod 
m,  and  a  be  prime  to  m,  then  art$  •••,ar^m)  is  also  a  reduced  resi- 
due system,  mod  m;  for  arlf  •••,  ar^m)  are  incongruent  each  to 
each,  mod  m,  prime  to  m  and  cf>(m)  in  number. 

Ex.     Since 

—  9,  2,  —17,  14,  15,  —4,  —13,  8,  19,  20 
constitute  a  complete  residue  system,  mod  10,  and  3  is  prime  to  10, 

—  27,  6,  —51,  42,  45,  —12,  —39,  24,  57,  60 
is  also  a  complete  residue  system,  mod  10.      Likewise  since 

—  9,  —17,  —13,  19 
is  a  reduced  residue  system,  mod  10. 

—  27,  —51,  —39,  57 
is  also  a  reduced  residue  system,  mod  10. 

If  p  be  any  prime  number  and  a  any  integer  prime  to  p,  it  is 
evident  from  the  above  that  there  exists  an  integer  ax  such  that 

aaL=  1,  mod  p. 

We  call  ax  the  reciprocal  of  a,  mod  p. 


THE   RATIONAL   REALM CONGRUENCES.  57 

§  8.  Fermat's  Theorem  as  Generalized  by  Euler. 

Theorem  io.     If  m  be  any  rational  integer  and  a  any  rational 
integer  prime  to  m,  then  a0(w)  ss  i}  mod  m. 

Let  fit  ?2>   '  '  '}  r<b(nt)  i) 

be  a  reduced  residue  system,  mod  m.     Then  since 

afit  Of*  •  •  •,  arUm)  2) 

is  also  a  reduced  residue  system,  mod  m,  each  integer  of  2)  is 
congruent  to  some  integer  of  1),  mod  m,  that  is,  we  have 

ar1 

,  modw,  3) 


where  rf1>rj2>'",rj<f>,m)  are  the  integers  1),  though  perhaps  in  a 
different  order.  Since  t$vt$^  '">ruim  are  the  integers  1),  we 
have 

Multiplying  the  congruences  3)  together,  we  have 

a0(m)P  =  P,  modw,  4) 

where  P  is  prime  to  m,  since  each  of  its  factors  is  prime  to  m. 
Hence,  dividing  both  members  of  4)  by  P,  we  have 

o0(w)==i,  modw.  5) 

If  m==  ±  pn,  where  p  is  a  prime,  we  have 

ai\P\-D\P\^^  lf  mod/1,  6) 

and,  in  particular,  when  m  =  p 

abl-i  =  Ij  mod  p.  7)  ' 

If  p  be  positive,  7)  becomes 

aP-1^!,  mod/>;  8) 

that  is,  if  p  be  a  positive  prime  number,  and  a  an  integer  not  divis- 
ible by  p}  aP'1  —  1  is  divisible  by  p.  This  is  the  form  in  which 
the  theorem  was  enunciated  by  Fermat.1 

1  This  theorem  was  published  by  Fermat  in  1670,  without  proof.  Euler 
was  the  first  to  give  a  proof.  He  gave  two :  Comm.  Acad.  Petrop.  VIII, 
1741,  and  Comm.  Nov.  Acad.  Petrop.  VII,  p.  74,  1761. 


58  THE   RATIONAL   REALM CONGRUENCES. 

Ex.  i.    Let  m  =  i5;  a  =  2;  then  <f>(is)  =S. 

From  5)  it  follows  that 

2<f>(m)  =  2*  =  1,  mod  15 ; 
that  is,  256  =  1,  mod  15. 

Ex.  2.    Let  P  =  7;   a  =  —  3- 

From  7)  it  follows  that       (—  3)8=i,  mod  7; 

that  is,  729  =  1,  mod  7. 

Ex.  3.     Let 

m=zpn  =  f;  a  =  2;  then  <f> (3*)  =2-3  =  6. 

From  6)  it  follows  that 

26  be  1,  mod  9 ; 
that  is,  64==  1,  mod  9. 

On  account  of  the  great  importance  of  Fermat's  theorem,  we 
shall  give  for  the  form  8)  a  second  proof,  depending  upon  the 

binomial  theorem.     If      aP^==a,  mod/>,  9) 

where  p  is  a  positive  prime,  hold  for  every  integral  value  of  a, 

then  aP'1^  1,  modp 

holds  when  a  is  prime  to  p. 

We  shall  show  now  that  9)  holds  for  all  integral  values  of  a. 
We  see  that  9)  holds  when  a=i.  If,  therefore,  we  can  show 
that  a  sufficient  condition  that  9)  shall  hold  for  a  =  a1-\-i  is 
that  it  shall  hold  for  a  =  ox,  9)  will  hold  for  all  positive  integral 
values  of  a.     We  have  by  the  binomial  theorem 

(a  +  1  )p  =  op  +  pav~-  ^titzzl}^  +  . . .  +'<*-'>"*%  +  1. 

V        '        '  '    t  '  12  "l-2  '••   (/> i)       ' 

From  §  10  we  know  that  all  coefficients  in  this  expansion  are 
integers.  Hence  since  p  occurs  as  a  factor  in  the  numerator  of 
the  coefficient  of  every  term  except  the  first  and  last,  and,  since 
the  denominators  of  these  terms  contain  only  factors  that  are 
prime  to  p,  the  coefficient  of  every  term  except  the  first  and  last 
is  divisible  by  p,  and  we  have 

(a4-i)P  =  aP+i,  modp, 
for  every  integral  value  of  a. 


THE   RATIONAL   REALM CONGRUENCES.  59 

Therefore  ((%  -f  I  )p  =  of  +  i,  mod  p, 

whence  assuming  that  9)  holds  for  a=.a1;  that  is, 

0/  =  ^,  mo&p, 

we  have  (a±  +  i)p^ax  +  1,  mod/) ; 

that  is,  9)  holds  for  a  =  a1-{-  1,  if  it  holds  for  a  =  at.  But  9) 
holds  for  0=1.  Hence  9)  holds  for  every  positive  integral 
value  of  a.  Moreover,  since  every  negative  integer  is  congruent 
to  some  positive  integer,  mod  p,  9)  holds  also  for  all  negative 
integral  values  of  a. 

Fermat's  theorem  in  the  form  8)  is  an  immediate  consequence 
of  the  theorem  that  we  have  just  proved. 

§  9.    Congruences  of  Condition.    Preliminary  Discussion. 

The  congruences  which  we  have  so  far  considered  may  be  com- 
pared to  arithmetical  equalities,  the  values  of  the  quantities  in- 
volved being  given  and  the  congruence  simply  expressing  the  fact 
that  the  difference  of  the  two  numbers  is  divisible  by  the  modulus. 

We  shall  now  consider  congruences  which  hold  only  when 
special  values  are  given  to  certain  of  the  quantities  involved ;  that 
is,  the  values  of  these  "  unknown  "  quantities  are  determined  by 
the  condition  imposed  by  the  congruence;  for  example,  let  x  be 
determined  by  the  condition  that  its  square  is  to  be  congruent  to 

2,  mod  7.     We  have  x2=z 2,  mod  7, 

and  see  easily  that  we  must  have 

xmz  or  —3,  mod  7/ 

To  develop  the  theory  of  congruences  of  condition,  it  is  neces- 
sary to  introduce  the  conception  of  the  congruence  of  two  poly- 
nomials with  respect  to  a  given  modulus;  thus,  if  f{xlfx2,  •••,*») 
be  a  polynomial1  in  the  undetermined  quantities  x1,x2,  --,xn  with 
rational  integral  coefficients,  we  say  that  f(xlfx2,  •••,#„)  is  iden- 
tically congruent  to  0  with  respect  to  the  modulus  m,  if  all  its 
coefficients  be  divisible  by  m. 

1  We  shall  understand  by  a  polynomial  in  n  undetermined  quantities 
Xi,x2,  ••-,xn  a  rational  integral  function  of  Xitx2,  ••-,xn  whose  coefficients, 
unless  the  contrary  be  expressly  stated,  are  rational  integers. 


60  THE   RATIONAL   REALM CONGRUENCES. 

This  relation  is  expressed  symbolically  by 

f(x1,xa,---,xH)=o,  modw.1 

Two  polynomials  f(xlyx2,  •••,  xn)  and  4>(xlfx2f  '-,xm)  are  said 
to  be  identically  congruent  to  each  other,  mod  m,  if  their  differ- 
ence  be  identically  congruent  to  o,  mod  m,  or  what  is  the  same 
thing  if  the  coefficients  of  corresponding  terms  in  the  two  poly- 
nomials be  congruent;  that  is,  in  symbols 

f(x1,x2,'~,xH)==<f>(x1,x2,~-,xn),  mod™, 

if  f(x1,x2,-.,xn)—<l>(x1,x2,--,xn)=o,  modw. 

For  example,  we  have 

8x2  —  2xy  -f-  6y  +  I  =  2x2  +  xy  —  2,  mod  3, 

since  6x2  —  3*3'  -f-6y  —  3=0,  mod  3, 

or,  in  other  words,  since 

8  =  2,  — 2=1,  6=0,  and  1=  —  2,  mod  3. 

If  f(xltx2,  •••', xn)  ^<f>(xlfx2y  '-',Xn),  mod  m,  and  alfa2,  ~,an 
be  any  n  integers,  then  evidently 

f(a1,a2,'--,an)===<l>(a1,a2,--,an).  modiw. 

If,  however,  all  the  coefficients  of  f(xlfx2,  •■-,xn)  be  not  congru- 
ent, mod  m,  to  the  corresponding  coefficients  of  <f>(xlfx2t-",xn), 
we  do  not  have  in  general 

f(alta2f  ••-,aw)=^(a1,a2,  •  -,o»),  mod  in,  1) 

for  ever>-  set  of  integers  olf  a2,  •  •  •,  a*.  The  demand  that  xx,  x2,  ■  •  •,  .r„ 
shall  have  such  values  and  only  such  that  1)  will  hold  is  expressed 
by  writing 

f{xXix2i  ~-,Xn)=4>(xlix2,---yXn),  mod  ;;/.  2) 

Any  set  of  integers  satisfying  1)  is  called  a  solution  of  2). 
The  determination  of  all  such  sets,  or  the  proof  that  none  exist, 
is  called  solving  the  congruence  2).  It  is  customary  to  say,  how- 
ever, that  a  congruence  is  solvable  or  unsolvable  according  as  it 
has  or  has  not  solutions.     We  call  2)  a  congruence  of  condition. 

1  The  symbol  ==  is  read  "  is  identically  congruent  to." 


THE   RATIONAL   REALM CONGRUENCES.  6  I 

If  alfa2,-",an  and  br,b2,---,bn  be  two  sets  of  n  rational  in- 
tegers and 


a2?=b2 


k  mod  m,  3) 


then  by  §  1,  v,    ■ 

f(alfa2,-",  a„)  =  f(blt  b2,  — ,V)a  mod  M, 

and  <^(a1?a2,  •••,o„)=<^(&1,  &2,  ■••,&,),  modw. 

Hence,  if  ax,  a2,  •  •  •,  a„  be  a  solution  of  2),  blf  b2l  •••,&»  is  also  a 
solution.  Two  solutions  so  related  are,  however,  looked  upon  as 
identical. 

In  order  that  two  solutions  may  be  counted  as  different,  it  is 
necessary  and  sufficient  that  there  shall  be  in  the  one  solution  a 
value  of  at  least  one  unknown  which  is  incongruent,  mod  tn,  to 
the  value  of  the  same  unknown  in  the  other  solution ;  that  is,  the 
n  relations  3)  must  not  hold  simultaneously. 

It  is  evident  from  the  above  that  in  order  to  solve  any  con- 
gruence, as  2),  it  is  sufficient  to  substitute  for  the  unknowns  the 
\tn\n  sets  of  values  obtained  by  putting  for  each  unknown  the  \tn\ 
numbers  of  a  complete  residue  system,  mod  m,  and  observe  which 
values  of  f(xlfx2,  -",xn)  so  obtained  are  congruent  to  the  corre- 
sponding values  of  <f>(xlfx2J  •••,.r„),  mod  m.  There  being  only  a 
finite  number,  |w|",  of  possible  solutions,  we  can  by  this  process 
always  completely  solve  any  given  congruence.  If  the  congruence 
have  the  form 

f(xx,x2,  •••,jt„)=o,  mod  mi, 

and  alfa2,  "-9Om  be  a  solution,  then  f{xx,x2,  •••ixn)  is  said  to  be 
zero,  mod  m,  for  these  values  of  xx,x2,  '-'fxn. 

Ex.    Let  us  consider  the  congruence 

f(x,y)  =2x*  —  xy  +  y  —  2f  +  1=0,  mod  3.1  4; 

1  In  order  to  avoid  confusion,  we  shall  use  throughout  this  book  the 
symbol  =  instead  of  as  to  denote  algebraic  identity. 


62  THE   RATIONAL   REALM CONGRUENCES. 

Putting  for  x  and  y,  the  numbers  —  i,  o,  I  of  a  complete  residue  system, 
mod  3,  we  obtain  nine  values  of  /  (x,  y). 

/(0,-l)=-2,      /(l,_l)=I,      /(_!,_!)=_!, 

/(o,  o)      =      i,    /(i,  o)      =3,    /(— i,  o)      =3, 
/(o,  i)      =o,         /(i,  i)      si,    /(-i,  i)      =3, 

Four  of  these  values  /(o,  i),  /(i,o),  /( — i,o),  and  /( — i,  i)    are  con- 
gruent to  o,  mod  3.      Hence  the  solutions  of  4)  are: 


.r=      1,    y==o 
x==—i,    y==ot 


mod  3. 


By  the  degree  of  a  polynomial,  mod  m,  we  shall  understand  the 
degree  of  the  term,  or  terms,  of  highest  degree,  whose  coefficient, 
or  coefficients,  are  not  divisible  by  m. 

A  reduced  polynomial,  mod  p,  is  one  whose  coefficients  are  all 
numbers  of  the  residue  system,  0,1,  •••,/> —  1. 

§  10.     Equivalent  Congruences. 

Addition  and  Multiplication  Transformations.   Two  congruences 

A Oi,  *%* ' "4  ** ) '■■  f»(f 1* *%,  ••'*,  **) j  mod  m,  1 ) 

and  <t>1(x1,x2,---,xn)=<f>2(x1,x2,'-',xn),modm,  2) 

are  said  to  be  equivalent  when  every  solution  of  the  first  is  a  solu- 
tion of  the  second,  and  every  solution  of  the  second  is  a  solution 
of  the  first. 

In  solving  a  congruence,  as  in  the  case  of  algebraic  equations, 
we  proceed  under  the  assumption  that  a  solution  exists  and  look 
upon  the  congruence  as  an  identity  in  the  values  of  x1}x2,  --,xn 
that  satisfy  it,  though  as  yet  unknown.  Looking  then  upon  1) 
as  an  identity  in  these  unknown  values  of  xx,x2,  --,xn,  we  con- 
sider what  operations  can  be  performed  upon  1 )  that  will  produce 
another  identity  2)  such  that  each  of  these  identities  is  a  neces- 
sary consequence  of  the  other.  Operations  of  which  this  is  true 
we  shall  call  reversible  operations. 

Referring  to  §1,  we  see  that  there  are  two  such  operations: 
first,  if  1 )  be  the  given  congruence  and 

Fx(x1,x2,--,xn)=F2(x1,x2,'-'ixn),  modw,  3) 


THE   RATIONAL   REALM CONGRUENCES.  63 

be  any  identical  congruence,  mod  m,  in  xlyx2,  -",xn,  we  can  add 
3)  member  by  member  to  i),  obtaining 

+  Fs(x1,xt,—fxn),  modw, 

a  congruence  equivalent  to  i). 

By  means  of  this  transformation,  we  can  transpose  any  term 
with  its  sign  changed  from  one  member  of  a  congruence  to  the 
other,  and  can  thus  reduce  any  congruence,  as  i),  to  an  equiva- 
lent congruence  of  the  form 

f(xlfx2,  '■•,x„)==o,  modm,  4) 

whose  second  member  is  o.     We  shall  hereafter  assume  the  con- 
gruences with  which  we  deal  to  have  been  reduced  to  this  form. 

We  may  also  by  this  transformation  reduce  the  coefficients  of 
f(xlfx2,  '-',Xn)  to  their  smallest  possible  absolute  values,  mod  m, 
and  thus  lessen  the  labor  of  solving  the  congruence. 
Ex.    The  congruence 

14X4 — wx3  -f-  zx2  +  7X  —  12  ==0,  mod  7,  5) 

is  equivalent  to  the  congruence 

—  3**  -f-  2X2  +  2  ==  o,  mod  7, 

which  has  two  roots  x  a  —  1  or  2,  mod  7,  and  these  are  therefore  the 
roots  of  5). 

A  second  operation  which,  when  performed  upon  any  congru- 
ence, as  1)  or  4),  yields  an  equivalent  congruence,  is  the  multipli- 
cation of  both  members  of  the  congruence  by  any  integer,  a,  prime 
to  the  modulus ;  that  is,  the  congruences 

f(xx, x2,---,xn)==B o,  mod m, 
and  &f(xv x2>  '">xn)  =0,  mod m, 

where  a  is  prime  to  m,  are  equivalent. 

Conversely,  we  may  divide  all  the  coefficients  of  a  congruence 
by  any  integer  prime  to  the  modulus,  obtaining  an  equivalent 
congruence. 

Ex.    The  congruences 

I5^y  —  2ixy  -f  3y2  +  9  ==  0,  mod  35 


64  THE   RATIONAL  REALM CONGRUENCES. 

and  S^y  —  7xy  +  y2  +  3  =  0,  mod  35 

are  equivalent. 

As  a  special  case  of  the  multiplication  transformation,  as  we 
shall  call  the  second  of  the  above  transformations,  we  have  the 
multiplication  of  the  congruence 

f(x1,x2,---,xn)=o,  modm, 

by  —  1 ;  that  is,  the  change  of  sign  of  each  of  its  coefficients. 

§  11.     Systems  of  Congruences.1    Equivalent  Systems. 

So  far  we  have  considered  only  single  congruences ;  that  is,  the 
unknown  quantities  are  subjected  to  a  single  condition.  We  can, 
however,  as  in  the  case  of  algebraic  equations,  subject  them  to 
two  or  more  conditions  simultaneously;  that  is,  xltx±i  "*,xn  may 
be  required  to  satisfy  simultaneously  the  congruences 

fi(x1,x2J---,xn)^o,  modWi, 

ft Oi,  *2>  * ' '>  *n)  =  o,  mod  m2, 


fr(*is  *»••'•  *j  *n)  s=  o,  mod  mr. 

By  a  solution  of  such  a  system  of  congruences  we  understand 
a  set  of  values  of  x\,x2,'-',xn  which  satisfy  simultaneously  all 
the  congruences. 

Two  solutions,  alya2,  ~-yan  and  blt  b2,  •••,&«,  are  considered  dif- 
ferent when  and  only  when  the  nr  congruences 


a2  =  b2 


^,mod  m&  mod  m2,  ••-,  mod m, 


an=bn~ 
are  not  satisfied  simultaneously. 

Two  systems  of  congruences  are  said  to  be  equivalent  when 
each  solution  of  the  first  system  is  a  solution  of  the  second  and 
each  solution  of  the  second  is  a  solution  of  the  first.  It  is  evident 
that  any  one  of  the  congruences  of  the  system  can  be  transformed 

1  See  Stieltjes:    Essai  sur  la  theorie  des  Nombres. 


THE    RATIONAL    REALM CONGRUENCES.  65 

into  an  equivalent  congruence  by  the  transformations  of  the  last 
article  and  the  system  so  obtained  will  be  equivalent  to  the  origi- 
nal system.  If  the  moduli  be  the  same,  we  can  obtain  an  equiva- 
lent system  by  adding  two  congruences  and  taking  the  new  con- 
gruence together  with  the  r  —  2  of  the  original  ones  not  used  and 
either  one  of  those  used.     Thus  the  system 

/iC*if*i»  •••»•*•)  ■B*i  modw,  )  x 

is  equivalent  to  the  system 

fx  C*u  *»'•"> x*  )  =  °>  mod  m, 
f1(x1,x2,  ■■-,xn)  +f%{x*>x9,—,*%)  =0,  modw, 

or,  more  generally,  if  ax,a2  be  any  two  integers  prime  to  m,  1)  is 
equivalent  to  the  system 

fx(xx,x2,  •••,*„)  ==0,  modw, 

oJi(xltx2t  ■■•,xn)  -\-a2f2(xlfx2,  •••,#„)  =0,  modw. 

Ex.    Let  the  given  system  be 

4-^  —  3?  +  7*  =  5  ] 

$x  +   y  —  3z==2    L  mod  17.  2) 

x  —  43/  —  s==i  J 

Multiplying  the  third  congruence  first  by  — 4  and  then  by  — 5,  and 
adding  it  to  the  first  and  second  respectively,  we  obtain  the  system 

I3y  +  ii2=      il 

2iy-j-   2z  =  —  3  Lmodi7,  3) 

'  *  —  43?  —     2  ==      1  J 
that  is  equivalent  to  2). 

Adding  the  first  and  second  congruences  of  3),  we  obtain  the  equiva- 
lent system 

132  =  — 2I 
2iy-f  22  =  —  3  L  mod  17. 
x  —  4y—   z=      1  J 

The  congruence  132^  —  2,  mod  17, 

has  the  single  solution  z  ==  —  8,  mod  17, 

that  substituted  in  2iy-|-2.s== —  3,  mod  17, 

gives  yas — 1,  mod  17. 

Substituting  these  values  of  y  and  8  in 

#  —  4y  —  z  ;=  1,  mod  17, 
5 


66  THE   RATIONAL   REALM CONGRUENCES. 

we  have  x^6,  mod  17. 

We  obtain  therefore  as  a  solution  of  the  given  system 

x  ==  6,   y  ==  —  1,   8  ess  —  8,   mod  17, 

a  result  easily  verified  by  substitution  in  the  original  system.  The  method 
of  solution  shows  that  this  is  the  only  solution  (see  §13). 

§  12.  Congruences  in  One  Unknown.  Comparison  with 
Equations. 

The  general  congruence  in  one  unknown  has  the  form 

/( x)  —  a0xn  +  &xxn-x  +  •  •  •  +  an  =  o,  mod  m.  1 ) 

If  r  be  a  rational  integer  such  that 

f(r)  =0,  modm, 
r  is  called  a  root  of  1). 

The  degree  of  1)  is,  as  has  been  said,  the  degree  of  the  term 
of  highest  degree  whose  coefficient  is  not  divisible  by  m. 

Such  a  congruence  presents  many  analogies  to  the  equation 

a0xn  +  axxn^  +  •••  +  <zn  =  o;  2) 

for  example,  to  the  addition  to  both  members  of  the  equation  of 
the  same  function  of  the  unknown  corresponds  the  addition  to 
the  members  of  the  congruence  of  any  functions  of  the  unknown 
which  are  identically  congruent  with  respect  to  the  modulus,  and 
to  the  multiplication  of  the  equation  by  any  quantity  not  a  func- 
tion of  the  unknown  corresponds  the  multiplication  of  the  con- 
gruence by  any  integer  prime  to  the  modulus. 

If  m  be  a  prime  number  the  congruence  presents  still  other 
striking  analogies  with  algebraic  equations,  these  analogies  being 
absent  in  the  case  of  a  composite  modulus. 

For  example,  consider  the  two  congruences  of  the  second 
degree 

O—  i)0  —  3)  so,  mod  7,  3) 

and  (x — i)(.r  — 3)  =0,  mod  12.  4) 

We  see  that  3)  has  two  roots,  1  and  3,  while  4)  has  four  roots, 
1,  3,  7  and  9;  that  is,  3)  has  a  number  of  roots  equal  to  its  degree, 
while  4)  has  more  roots  than  its  degree. 

The  analogy  with  algebraic  equations  in  the  case  of  the  prime 


THE   RATIONAL   REALM CONGRUENCES.  67 

modulus  is  as  evident  as  is  the  lack  of  analogy  in  the  case  of  the 
composite  modulus.  We  shall  see  later  that  no  congruence  of  the 
form  1)  with  prime  modulus  can  have  more  roots  than  its  degree. 

The  reason  for  this  difference  in  the  case  of  the  above  example 
is  seen  to  be  that,  if  a  be  any  integer,  the  product  (a —  i)  (a  —  3) 
is  divisible  by  a  prime  number,  as  7,  when  and  only  when  one  of 
its  factors  is  divisible  by  this  prime,  a  statement  no  longer  true 
when  the  modulus  is  composite ;  that  is,  a  product  is  zero,  mod  m, 
when  and  only  when  one  of  its  factors  is  zero,  mod  m,  if  m  be  a 
prime  number,  but  not  otherwise.  We  shall,  therefore,  in  the 
discussion  of  the  general  congruence  of  the  form  1 )  confine  our- 
selves first  to  the  case  in  which  the  modulus  is  a  prime  and  shall 
then  show  that  the  solution  of  any  congruence  of  the  form  1) 
with  composite  modulus  can  be  reduced  to  the  solution  of  a  series 
of  congruences  of  the  same  form  with  prime  moduli. 

Although  striking  analogies  between  congruences  and  algebraic 
equations  have  already  been  pointed  out,  while  others  will  be 
observed  later,  it  is  important  to  note  an  essential  difference 
between  them. 

In  the  case  of  an  algebraic  equation  it  is  the  same  thing  to 
say  that  all  the  coefficients  of  an  equation  are  zero  or  that  it  is 
satisfied  by  every  value  of  the  unknown  quantity,  each  of  these 
properties  implying  the  other. 

In  the  case  of  congruences,  however,  although,  if  the  coefficients 
be  all  congruent  to  zero  with  respect  to  the  modulus,  the  con- 
gruence is,  of  course,  satisfied  by  any  integral  value  of  the 
unknown,  on  the  other  hand,  it  is  not  true  in  general  that,  if  a 
congruence  be  satisfied  by  all  integral  values  of  the  unknown,  that 
all  of  its  coefficients  are  divisible* by  the  modulus. 

For  example,  as  is  easily  seen  from  Fermat's  theorem,  the 
congruence 

xp  —  .r==o,  modp, 

where  p  is  a  prime,  is  satisfied  by  every  integral  value  of  x;  but 
its  coefficients  are  not  all  divisible  by  p.  The  reason  for  the  dif- 
ference will  be  shown  later.     We  shall  see  also  that,  although  a 


68  THE   RATIONAL   REALM CONGRUENCES. 

congruence  of  the  form  i )  with  prime  modulus  can  not  have  more 
roots  than  its  degree,  it  can  have  less;  for  example,  the  three 
congruences 

x3  —  2x2  —   x  -\-  2  ==  o,  mod  5, 

x3  +  2*2  —  2x  -\-  i  =3  o,  mod  5, 

x3  +  4X2  +    x  -f- 1 33=  o,  mod  5, 

that  are  all  of  the  third  degree  and  have  the  same  prime  modulus, 
5,  have  respectively  three  roots,  1,  —  1,  and  2,  one  root,  — 2,  and 
no  root. 

Before  taking  up  the  general  congruence  in  one  unknown,  we 
shall  consider  that  of  the  first  degree. 

§  13.    Congruences  of  the  First  Degree  in  One  Unknown. 

The  most  general  congruence  of  the  first  degree  can  be  written 
in  the  form 

ax^=b,  mod  m. 

We  shall  consider  first  the  case  where  a  is  prime  to  m. 
Theorem  ii.     The  congruence 

\  ax==b,  modra, 

where  a  is  prime  to  m,  has  always  one  and  but  one  root. 

If  we  put  for  x  successively  the  \m\  integers  mlfm2,  '••,mm  of 
a  complete  residue  system,  mod  m,  we  obtain  \m\  integers  amlf  am2, 
~-,amm,  that  also -constitute  a  complete  residue  system  (Th.  9), 
and  it  is  evident  that  one  and  but  one  of  these  integers,  say  ami, 
will  be  congruent  to  b,  mod  m.  Hence  the  congruence  has  always 
one  and  but  one  root,  Wj.  We  can  evidently  solve  any  congru- 
ence of  this  form  by  this  method. 

Ex.     Let  the  given  congruence  be 

3*3=  — 5>  mod  14.  1) 

Taking  as  a  complete  residue  system,  mod  14,  the  integers  o,  1,  2,  3,  •••,  13, 
and  putting  x  equal  to  these  values  in  succession,  we  have 

$x  =  6,  3,  6,  9,  12,  15,   18,  21,  24,  27,  30,  33,  36,  39. 

The  only  one  of  these  integers  that  is  congruent  to  —  5,  mod  14,  is  9 ;  that  is, 

3-3  33=  — 5,  mod  14. 

Hence  ^==3,  mod  14,  is  the  single  root  of  1) 


THE   RATIONAL   REALM CONGRUENCES.  69 

By  means  of  Fermat's  theorem  we  can  find  a  general  expres- 
sion for  the  root  of  a  congruence  of  the  above  form. 
Since  a  is  prime  to  m,  we  have 

a0(m)  ==  I}  modm, 

which  multiplied  by  b  gives 

ba<t>(m}^b,  modm, 

or  aba(P(m)-1^b,  modm. 

Hence  £?a0(m)_1  is  the  root  of  the  congruence 

ax^b,  modm, 
where  a  is  prime  to  m. 

This  is  one  of  the  few  cases  in  the  theory  of  numbers  where  the 
quantity  sought  can  be  expressed  as  an  explicit  function  of  the  given 
quantities. 

Ex.    The  root  of 

3*==  — 5,  mod  14, 
is  jt==  —  5-3*(14)_1,  mod  14; 

that  is,  *===  —  5-35== — n==3,  mod  14. 

We  shall  now  consider  the  general  case  where  a  is  any  integer 
that  may  or  may  not  be  prime  to  m. 

Theorem  12.  The  necessary  and  sufficient  condition  for  the 
solvability  of  the  congruence 

ax^E=b,  modm, 

is  that  b  shall  be  divisible  by  the  greatest  common  divisor,  d,  of  a 
and  m,  and  when  this  condition  is  fulfilled,  the  congruence  has 
exactly  \d\  incongruent  roots. 

Let  a  =  axd  and  m  =  mxd,  where  ax  is  prime  to  mx.     From 

ax==b,  modm,  2) 

we  have  axdx  =  b  +  kmx d. 

Hence  b  must  be  divisible  by  d;  that  is,  b  =  bxd  is  a  necessary 
condition  that  2)  can  be  solved.     This  gives 

axdx  =  bxd  -\-  kmxd,  3) 

or  axx  =  bx,  modm^  4) 


yO  THE   RATIONAL   REALM CONGRUENCES. 

Since  ax  is  prime  to  mx,  4)  has  a  root  (Th.  11).  Moreover, 
all  roots  of  4)  are  also  roots  of  2)  ;  for  from  4)  follows  3)  and 
hence  2).  Therefore  the  divisibility  of  b  by  d  is  a  sufficient  as 
well  as  necessary  condition  for  the  solvability  of  2).  We  see  also 
that  not  only  are  all  roots  of  4)  roots  of  2),  but  all  roots  of  2) 
satisfy  4)  and  are  therefore  integers  of  the  form  r  +  kmx,  where 
r  is  a  root  of  4).  We  ask  now  how  many  of  these  roots  are  in- 
congruent  to  each  other,  mod  m;  that  is,  how  many  incongruent 
roots  has  2)  ?  Any  two  roots,  r  +  k±mv  r  +  k2mx,  of  4)  are  con- 
gruent, mod  m,  when  and  only  when 

r  -\-  k1mx  —  (r  -f-  ^wj  —  nm, 

where  n  is  an  integer ;  that  is,  if 

(  kx  —  k2 )  m1  —  nmxd, 

or  kx  —  k2  =  nd, 

or  kx^k2,  modd. 

Hence,  in  order  that  the  roots  of  2)  shall  be  incongruent,  it  is 
necessary  and  sufficient  that  the  values  of  k  shall  be  incongruent, 
mod  d.  If  we  put,  therefore,  for  k  the  \d\  integers  of  a  complete 
residue  system,  mod  d,  for  example,  o,  1,2,  •  •  •,  \d\  —  1,  we  shall 
obtain  all  the  incongruent  roots  of  2),  namely 

r,  r  +  mt,  r  +  2wx,  •••,r+(|d|  —  1)  mx. 

They  are  evidently  \d\  in  number. 

Ex.     Consider  the  congruence 

1 2.x  ^  —  20,  mod  56.  5 ) 

Here  d  —  4.     Dividing  by  4  we  have 

3*  =  —  5,  mod  14, 

a  congruence  whose  root  has  already  been  found  to  be  — 11.  Therefore 
the  roots  of  5)  have  the  form  — 11 -{- 14&,  and  are  four  in  number. 
They  are  — 11,  3,  17  and  31. 

§  14.  Determination  of  an  integer  that  has  certain  residues 
with  respect  to  a  given  series  of  moduli. 

Let  us  consider  first  the  case  in  which  the  required  integer  has 
to  satisfy  two  such  conditions ;  that  is,  we  are  to  determine  x  so 
that  we  have  simultaneously 


THE   RATIONAL   REALM CONGRUENCES.  7  I 

x  =  ax,  modwu  i) 

and  x  =  a2,  modm2.  2) 

All  integers  satisfying  1)  have  the  form  x  =  ax-\-mxy,  where  y 
is  an  integer.     Since  x  must  also  satisfy  2),  y  must  satisfy  the 

condition  mxy  =  a2 —  ax,  mod  w,.  3) 

By  Th.  12  for  3)  to  have  a  solution,  it  is  necessary  and  sufficient 
that  a2  —  ax  shall  be  divisible  by  the  greatest  common  divisor,  d, 
of  mx  and  m2.  If  this  requirement  be  fulfilled  and  y0  be  one 
root  of  3),  every  root,  y,  of  3)  must  satisfy  the  condition 

that  IS,  y=yQJr-jyv 

where  yx  is  any  integer.  All  integers  satisfying  both  1)  and  2) 
have  therefore  the  form 

mm 
x  =ax  +  mxy0  +  —^yx  \ 

.       .  .,  m,m„ 

that  is,  x  =  ax  +  mxy0,  mod     *      . 

Hence  if  x0  be  any  integer  satisfying  both  1)  and  2),  all  and  only 
those  integers  satisfy  both  1)  and  2)  that  are  congruent  to  x0  with 
respect  to  the  least  common  multiple  of  the  moduli  of  1)  and  2). 
By  an  easy  extension  of  this  method  we  obtain  the  common 
solution,  if  any  exist,  of  the  n  congruences 


x==ax,  modwj, 
x==a2,  mod  w2, 


4) 


x^=an,  mod  ///„, 

and  we  see  that,  if  x0  be  an  integer  satisfying  all  these  congru- 
ences and  /  the  least  common  multiple  of  the  moduli, 

.r  ==  x0,  mod  /, 

gives  all  the  common  solutions  of  the  system  4).     The  general 


72  THE   RATIONAL   REALM— CONGRUENCES. 

problem  of  determining  whether  any  given  system  of  congruences 
of  the  form  ax^=b,  mod  m,  have  common  solutions  and  of  find- 
ing them,  if  they  exist,  can  be  solved  by  the  above  method.  When 
the  coefficients  of  x  are  prime  to  the  moduli  the  congruences  can 
evidently  be  reduced  to  the  form  x=c,  mod  m,  and  we  have  the 
case  just  treated.     If  the  moduli  be  prime  each  to  each, 

/  =  m1m2  •"  mn 

and  the  congruences  4)  always  have  a  common  solution. 

We  shall  now  give  another  solution  of  this  problem  for  the 
special  case  last  mentioned.  This  solution  is  interesting  on  ac- 
count both  of  its  symmetry  and  some  important  deductions  that 
can  be  made  from  its  form.  W7e  have  then  to  determine  the 
common  solutions  of  the  congruences  4) ,  the  moduli  mlf  m2,  •-,  mn 
being  prime  each  to  each. 

We  determine  first  for  each  modulus,  mi,  an^auxiliary  integer, 
bi,  such  that  bi  is  congruent  to  1  with  respect  to  the  modulus  nti 
and  is  divisible  by  each  of  the  other  moduli,  and  hence  by  their 
product ;  that  is,  we  determine  blf  b2,--,bn  so  that 

bx=i,  modm^  and  ^  =  0,  mod m2m3  •••#»«, 
&2==i,  mod  w2,  and  b2  =  o,  mod  m1ms  •  •  •  mn, 


bn=i,  modmn,  and  bn=o,  mod m1m2  •  •  •  mn_^. 

It  is  evident  that  this  can  always  be  done,  for  we  have  in  the  case 
of  bx  from  the  second  condition  b1  =  tn2m3--  mnclf  and  it  only 
remains  to  determine  a  value  for  ct  in  accordance  with  the 
condition 

m2mz  -  -  -  mnc1  mm  1,  mod  mlf 

that  is  always  possible  since  m2ms  •  •  •  tjtn  is  prime  to  mx. 
Having  found  these  auxiliary  integers,  we  put 

r  =  a1b1  +  a2b2  -f  •••  +  anbn, 


THE   RATIONAL   REALM CONGRUENCES.  73 

and  shall  show  that  the  common  solutions  of  4)  are  the  integers 
satisfying  the  congruence 

jrasr,  mod  mxm2  •  •  •  m„.  5) 

If  x  satisfy  5),  then 

jir  =  r,  mod  mi,  6) 

and,  since  all  the  auxiliary  integers  except  bi  are  divisible  by  mi, 
from  6)  it  follows  that 

x  =  ciibi,  mod  Mi, 

and  hence,  since  bi^i,  mod m^, 

we  see  that  .r==ai,  mod  m^ 

Hence  every  integer,  that  satifies  5),  satisfies  each  of  the  con- 
gruences 4).  Moreover,  every  integer,  that  satisfies  each  of  the 
congruences  4),  satisfies  5),  for,  if  x0  be  such  an  integer,  then 
from 

x0==cii,  mod nii, 

and  r  =  di,  modwj, 

we  see  that  x0  —  r  =  o,  modWij 

that  is,  x0  —  r  is  divisible  by  each  one  of  the  moduli  mlf  m2,  •  •  •,  tnn, 
and  hence,  since  they  are  prime  to  each  other,  by  their  product. 

Therefore  x0==r,  mod  m1m2  •••  mn.  Hence  the  integers  satis- 
fying 5)  are  all  the  common  solutions  of  4).  It  will  be  observed 
that  the  auxiliary  integers  bx,b2,-",bn  are  entirely  independent 
of  a1}  a2,  ■••,an,  being  dependent  only  on  the  moduli. 

Ex.  It  is  required  to  find  the  common  solutions  of  the  congruences 
x==2,  mod  11,    x===4,  mod  15,    x==9,  mod  14. 

To  calculate  the  auxiliary  integers  bi,  b2,  b3,  we  have 

&x  =  2ioci==i,  mod  11, 
fr2=i54c2=i,  mod  15, 
b3  =  165^3^  1,  mod  14, 
and  hence  Ci==i,  modn,    ^  =  210, 

c2^ 4,  mod  15,    £2  =  6i6, 
cz^g,  modn,    &3=i485. 
Therefore  r  =  420  +  2464  +  13365  =  16249, 


74  THE   RATIONAL   REALM CONGRUENCES. 

whence  ;r==  16249,  11^2310^ 

or  x^yg,  mod  2310, 

a  result  that  is  easily  verified. 

We  observe  now  two  important  facts  concerning  r,  that  are 
direct  consequences  of  the  symmetrical  method  of  its  formation. 

First,  if  for  alt  a2,---,an  be  put  the  integers  of  complete  residue 
systems  with  respect  to  the  moduli  m1,m2,  •••,*»»,  respectively,  the 
resulting  values  of  r  form  a  complete  residue  system,  mod  /,  for 
we  obtain  thus  |/|  values  of  r  and  they  are  incongruent  each  to 
each,  mod  /.     To  show  this,  let  two  values  of  r  be 

r'  as  a1fb1  +  a%%  +  *  •  •  +  CLn'bn, 
and  r"  =  ax"bx  +  a2"b2  +  ■  ■  •  +  *•%* 

where  we  do  not  have  simultaneously 

ax'  ss  a/',  mod  mlf  a2'  =  a2",  mod  m2,  ■••,  a*'ss  an",  mod  mn ; 

that  is,  in  order  that  the  two  values  of  r  be  different  we  must 
have  at  least  one  of  the  a"s,  such  as  a/,  in  r'  incongruent,  mod  mi, 
to  the  corresponding  a",  a/',  in  r". 

Let  at  sfs  a" ,  mod  m%. 

If  r'  =  r",  mod/, 

it  would  follow  that       r'  ^  r",  mod  Wi, 
and  hence  also  ai'bi^=ai"b{, mod w», 

or,  since  5i=i,modw{, 

ai'^ai",  mod  Wi, 

that  is  contrary  to  our  supposition.  The  two  values  of  r  are 
therefore  incongruent  with  respect  to  the  modulus  /. 

In  the  second  place,  if  we  select  from  the  system  of  values  of  r 
just  formed  those  which  are  formed  by  putting  for  alfa2,---,a„, 
the  integers  of  reduced  residue  systems  with  respect  to  the 
moduli  mlfmv  -••,tnn  respectively,  the  resulting  values  of  r  form 
a  reduced  residue  system,  mod  /.  We  have  already  shown  that 
these  values  of  r  are  incongruent  each  to  each,  mod  /.  It  re- 
mains to  be  shown  that  all  and  only  those  values  of  r  that  are 
prime  to  /  occur  in  the  system  as  formed.     If  one  of  these  values 


THE   RATIONAL   REALM CONGRUENCES.  75 

of  r,  as  r',  =a1fb1  +  •  •  •  +  an'bn,  have  a  prime  factor,  p,  in  com- 
mon with  I,  then  some  one  of  the  moduli,  as  mi,  must  have  this 
factor  in  common  with  r,  and  since 

r'  =  ai,  mod  Mi, 
ai  and  nti  would  have  the  common  factor  p,  which  is  contrary  to 
the  hypothesis  that  «#'  is  an  integer  of  a  reduced  residue  system, 
mod  mi. 

Hence  all  values  of  r  obtained  above  are  prime  to  /.  More- 
over, when  a  value  of  r,  as  r',  is  prime  to  /,  a/,  a2,  '-,an  are  each 
prime  to  their  respective  moduli,  for,  if  any  a,  as  a,-',  have  a  factor 
p  in  common  with  its  modulus,  then  since 

r'  =  a/,  mod  Wi, 

r'  would  have  the  factor  p  in  common  with  m\%  and  hence  with  /. 
Hence  all  values  of  r,  that  are  prime  to  /,  occur  in  the  above  sys- 
tem, and  it  is  therefore  a  reduced  residue  system,  mod  /. 

Ex.    Let  mi  =6,    m2  =  5, 

we  have  bx  =  5C1  &  i,  mod  6, 

and  b2  =  6c2^=i,  mods, 

whence  Cis=5,  mod  6, 

and  c2  ^  i,  mod  5. 

Then  bx  =  25,  and  b2  =  6, 

whence  r  =  2501  -)-  6a2. 

Putting  for  ai  the  values  1, 5  and  for  a2  the  values  1, 2, 3, 4,  that  is, 
the  integers  of  reduced  residue  systems,  mod  6,  mod  5,  respectively,  we 
have  for  the  resulting  values  of  r  31,  37,  43,  49,  131,  137,  143,  149,  that, 
being  all  prime  to  30  and  in  number  0(30),  =  8,  constitute  a  reduced  resi- 
due system,  mod  30. 

This  method  of  forming  a  reduced  residue  system  shows  us  at 
once  that  the  number  of  integers  in  such  a  system,  mod  m1m2  ■  •  ■  mn, 
where  m19 m2,  ••-, mn  are  prime  each  to  each,  is  equal  to  the  prod- 
uct of  the  numbers  of  the  integers  in  the  reduced  residue  systems 
for  each  of  the  moduli  tnlt  m2,  •  •  •,  mn. 

We  obtain  therefore  a  new  proof  of  Th.  4 ;  that  is,  that 

4>(mxm2  ■■■  mn)=cf,(m1)<l>(m2)  ■■■<f>(m»), 
where  mx,m2,  '",mn  are  prime  each  to  each. 


y6  THE  RATIONAL   REALM CONGRUENCES. 

We  shall  proceed  to  the  discussion  of  the  general  congruence 
of  the  wth  degree  in  one  unknown  with  prime  modulus  and  shall 
first  develop  briefly  the  theory  of  the  divisibility  of  polynomials 
with  respect  to  a  prime  modulus. 

§  15.  Divisibility  of  one  Polynomial  by  another  with  respect 
to  a  Prime  Modulus.    Common  Divisors.    Common  Multiples. 

If  p  be  any  rational  prime  number  we  have  the  following 
definition:  A  polynomial,  f(x),  is  said  to  be  divisible  with  respect 
to  the  modulus  p  by  a  polynomial  <f>(x)  when  there  exists  a  poly- 
nomial Q(x)  such  that 

f(x)mQ(x)*(x),  modp. 

We  say  that  <f>(x)  and  Q(x)  are  divisors  or  factors,  mod  p,  of 
f(x),  and  that  f(x)  is  a  multiple,  mod  p,  of  <f>(x)  and  Q(x). 
We  also  say  that  /(•*")  is  resolved,  mod  p,  into  the  factors  <f>(x) 
and  Q(x).  The  degree  of  a  polynomial,  mod  p,  is  the  degree  of 
the  term  of  highest  degree  whose  coefficient  is  not  divisible  by  p. 
The  sum  of  the  degrees  of  the  factors  of  f(x)  is  evidently  equal 
to  the  degree  of  f(x). 

Ex.     It  is  easily  seen  that 

x*  +  3xA  —  4*3  +  2=  (2X2  —  3)(3x*  —  *2+  1),  mod  5. 
Hence  23?  —  3  and  3xs  —  x2  -f-  1  are  divisors,  mod  5,  of  x5  +  3**  —  42*  +  2. 

We  have  as  direct  consequences  of  the  definition  of  divisibility : 

i.  //  f±(x)  be  a  multiple,  mod  p,  of  /2(^')  and  f2(x)  be  a  mul- 
tiple, mod  p,  of  f3(x),  then  f±{x)  is  a  multiple,  mod  p,  of  fs(x)f 
or  more  generally,  if  each  polynomial  of  the  series  f1(x),  f2(x), 
'">fn(x)  be  a  multiple,  mod  p,  of  the  one  immediately  following, 
then  each  polynomial  of  the  series  is  a  multiple,  mod  p,  of  all  that 
follow. 

ii.  //  ft(x)  and  f2(x)  be  multiples,  modsp,  of  f{x),  then 
f\{x)  +/2(-r)  and  fi_(x) — f2(x)  are  multiples,  mod  p,  of  f(x), 
or  more  generally,  if  fx(x)  and  f2{x)  be  multiples,  mod  p,  of 
f(x),  and  F1(x),F2(x)  be  any  two  polynomials,  then  F1(x)f1(x) 
-f- F2(x)f2(x)  is  a  multiple  of  f(x). 

If  two  or  more  polynomials  f1(x),f2(x),---,fn(x)  be  divis- 
ible, mod  p,  by  a  polynomial  <f>(x),  <f>(x)  is  said  to  be  a  common 


THE   RATIONAL    REALM CONGRUENCES.  JJ 

divisor,  mod  p,  of  fx(x),f2(x),  •••,/„(».  If  a  polynomial  f(x) 
be  a  multiple,  mod  p,  of  two  or  more  polynomials  <£i(-r),<k>(-l')> 
•■■,<f>„(x),  f(x)    is  said  to  be  a   common  multiple,  mod  p,  of 

§  1 6.  Unit  and  Associated  Polynomials  with  Respect  to  a 
Prime  Modulus.     Primary  Polynomials. 

We  ask  now  whether  there  exist  polynomials  that  with  respect 
to  a  modulus  p  divide  all  polynomials.  Evidently  those  have  this 
property  that  are  of  degree  o  and  are  ^  o,  mod  p ;  that  is,  the  ra- 
tional integers  not  divisible  by  p,  for  they  are  divisors,  mod  p,  of  I 
and  i  divides  every  polynomial.  Furthermore,  these  are  the  only 
polynomials  having  this  property,  for  no  polynomial,  f(x),  of 
degree  higher  than  the  oth  can  divide,  mod  p,  all  polynomials,  for 
it  can  not  divide  i,  since  then  the  sum  of  the  degrees  of  the 
divisor  and  the  quotient,  mod  p,  would  be  greater  than  o,  the 
degree  of  i. 

We  call  the  rational  integers,  excluding  those  divisible  by  p,  the 
unit  polynomials,  mod  p,  or  briefly,  units,  mod  p,  and  since  two 
polynomials  that  are  congruent,  mod  p,  are  considered  as  identical, 
we  can  take  as  the  units,  mod  p,  the  integers  of  any  reduced  res- 
idue system,  mod  p,  for  example,  I,  2,  •••,  \p\  —  i. 

Thus  the  unit  polynomials,  mod  7,  are  1,  2,  3,  4,  5,  6. 

Two  polynomials  which  differ  only  by  a  unit  factor,  mod  p,  are 
called  associated  polynomials  and  are  looked  upon  as  identical  in 
all  questions  of  divisibility,  mod  p. 

If  two  polynomials,  f1(x),  /2  (■*"),  are  eacn  associated,  mod  p, 
*  with  a  third  polynomial,  they  are  associated  with  each  other;  for  if 

f1(x)=af3(x),  modp,  1) 

and  f2(x)=bf3(x),  modp,  2) 

where  a  and  b  are  units,  mod  p,  then,  multiplying  2)  by  blf  the 
reciprocal,  mod  p,  of  b,  we  have 

btft00mf9(x),  modp, 
and  hence  from  1) 

f1(x)=ab1f2(x),  modp, 
where  ab1  is  a  unit,  mod  p. 


yS  THE   RATIONAL   REALM CONGRUENCES. 

Two  polynomials,  that  are  associated,  mod  p,  are  evidently  of 
the  same  degree  and  each  is  a  divisor,  mod  p,  of  the  other. 

Conversely,  if  two  polynomials  be  each  divisible,  mod  p,  by  the 
other,  they  are  associated. 

Two  polynomials  that  have  no  common  divisor,  mod  p,  other 
than  the  units  are  said  to  be  prime  to  each  other,  mod  p. 

Any  polynomial,  f(x),  has  \p\  —  I  associates,  mod  p.  Of  these 
one  and  only  one  has  as  the  coefficient  of  its  term  of  highest 
degree  i.  This  one  is  called  the  primary  associate,  mod  p,  of 
f(x).     For  example,  the  six  polynomials 

x3  -\-2x  — -3,    2x3  +  4X  —  6,     3^r3  +  6x  —  2, 

4x3+   x  —  5,     5^3  +  3.r—  1,     6x3  +  5x  —  4, 

are  associated,  mod  7,  and  x3  +  2x  —  3  is  the  primary  one. 

§17.  Prime  Polynomials  with  respect  to  a  Prime  Modulus. 
Determination  of  the  Prime  Polynomials,  mod  p,  of  any  Given 
Degree. 

A  polynomial  that  is  not  a  unit,  mod  p,  and  that  has  no  divisors, 
mod  p,  other  than  its  associates  and  the  units,  is  called  a  prime 
polynomial,  mod  p. 

If  it  has  divisors,  mod  p,  other  than  these  it  is  said  to  be  com- 
posite, mod  p. 

To  find  the  primary  prime  polynomials,  mod  3,  of  any  given 
degree  we  may  proceed  as  follows,  considering  all  polynomials 
to  be  reduced.  All  polynomials  of  the  first  degree  are  evidently 
prime.  Hence  primary  prime  polynomials  of  the  first  degree, 
mod  3,  are  three  in  number,  namely 

x,     x  -\-  1,     x  -f-2. 

The  reduced  primary  polynomials,  mod  3,  of  the  second  degree 
are  nine  in  number,  namely 

X2,  X2  +  X,  X2  +  2X, 

X2+I,      X2  +  X-\-l,      X2-\-2X+I, 
X2  +  2,      X2  +  X  +  2,      X2-^-2X-\-2. 

From  the  three  primary  polynomials  of  the  first  degree,  we 
can  form  the  six  composite  polynomials  of  the  second  degree 


THE   RATIONAL   REALM CONGRUENCES.  79 


.i-2  =  X2,  x  (x  +  I  )  =  x-  +    X, 

(x  +  i)2==x2  +  2x  +  i,  ".r(^  +  2)=^r2  +  2jr, 


mod  3. 


These  being  the  primary  composite,  polynomials,  mod  3,  of  the 
second  degree,  we  see  that 

X2+I,      X2  +  X  +  2,      X2  +  2X  +  2, 

are  the  primary  prime  polynomials,  mod  3,  of  the  second  degree. 

In  like  manner  we  see  that  there  are  nineteen  composite  poly- 
nomials of  the  third  degree,  mod  3,  and  hence  eight  prime  poly- 
nomials of  the  third  degree,  mod  3,  since  there  are  in  all  twenty- 
seven  reduced  primary  polynomials  of  the  third  degree,  mod  3. 

It  can  be  shown  that,  when  n  is  greater  than  1,  the  number  of 
prime  polynomials,  mod  p,  of  the  nth.  degree  is 

y  n  n  n 

-  (pn  —  2/^  -f  2/^2  —  2/^3  ^ )f 

where  qx,q2i<lz,    ">  are  the  different  prime  factors  of  n. 

This  expression  being  always  different  from  o,  it  follows  that 
there  exist  prime  polynomials,  mod  p,  of  any  given  degree.1 

§  18.  Division  of  one  Polynomial  by  Another  with  Respect 
to  a  Prime  Modulus. 

Theorem  13.  //  f(x)  be  any  polynomial  and  <f>(x)  be  any 
polynomial  not  identically  congruent  to  o,  mod  p,  there  exists  a 
polynomial  Q(x),  such  that  the  polynomial 

f(x)  —  Q(x)4>{x)==R{x),  modp,  1) 

is  of  lower  degree  than  <f>(x). 

The  operation  of  determining  the  polynomials  Q(x)  and  R(x) 
is  called  dividing  f(x)  by  <f>(x),  mod  p.  We  call  Q(x)  the  quo- 
tient, and  R(x)  the  remainder  in  the  division,  mod  p,  of  f(x)  by 
<f>(x).  We  shall  prove  the  existence  of  Q(x)  andi?(^r)  by  giving 
a  method  for  their  determination. 

1 H.  J.  S.  Smith :  p.  153.  Borel  et  Drach :  pp.  49,  50.  Bachmann : 
Niedere  Zahlentheorie,  pp.  372,  373. 


80  THE   RATIONAL   REALM CONGRUENCES. 

Let                  /(*)  =  a0xn  +  a,*"-1  +  •  •  •  +  on, 
4>  (x)  =  b0xm  +  hi**-*  -\ h  bm 

be  any  two  polynomials  and  let 

&0=j=o,  mod  p. 

We  shall  consider  first  the  case  in  which  b0  is  I,  and  shall  then 
show  that  the  general  case  can  be  reduced  to  this  one.  Since  b0 
is  i,  we  can  divide  f(x)  by  <f>(x)  as  in  ordinary  division  until  we 
get  a  remainder  R(x)  of  lower  degree  than  <f>(x),  the  quotient 
being  Q(x).     We  have  then 

from  which  follows  at  once  i). 

We  can  now  reduce  to  this  particular  case  the  general  case  in 
which  b0  has  any  value  not  divisible  by  p.  Let  c0  be  the  recip- 
rocal, mod  p,  of  b0 ;  then 

c0</>0)  ==</>i 0')>  rnodp,  2) 

where  <f>x(x)  is  a  polynomial  the  coefficient  of  whose  term  of 
highest  degree  is  1  when  reduced,  mod  p.  Dividing  f(x)  by 
<f)1(x)  as  above,  we  have 

f{x)^Q{x)4>1{x)+R{x),  modp, 

and  hence,  making  use  of  2), 

f(x)mc0Q(x)^(x)+R(x),  modp, 

where  c0Q(x)  andi^(^r)  are  the  quotient  and  remainder  required.1 
The  above  theorem  plays  the  same  role  in  the  theory  of  the 
divisibility  of  polynomials  with  respect  to  a  prime  modulus  that 
Th.  A  does  in  that  of  rational  integers. 

Ex.  Let  it  be  required  to  divide,  mod  7, 

f{x)  =5XS  —  2Xt-{-2XS  —  5X1i  +  2X+I, 

by  <f>(x)=3xB+    x*—$x  —2. 

1  See  also  Cahen :  p.  70,  Borel  et  Drach :  p.  33,  and  Bachmann :  Niedere 
Zahlentheorie,  p.  368,  concerning  the  division  of  one  polynomial  by  another 
with  respect  to  a  prime  modulus.. 


THE    RATIONAL   REALM CONGRUENCES.  8  I 

Since  5  is  the  reciprocal,  mod  7,  of  3,  we  have 

0i O)  =50 O)  =.r8  +  5*2  +  3*  —  3,  mod  7.  3) 

Dividing  /O)  by  0iO)  as  in  ordinary  algebraic  division,  we  have 

Sx5  —  2jt4  +  2X3  —  sxz  +  2xJri  —  (sx2  —  27X  +  122)  (V  +  5**  +  3*  —  3) 

=  —519^  —  445^  +  367, 
whence,  reducing  coefficients,  mod  7, 

5jt5  —  2*4  +  2X3  —  5^-2  +  2X  +  1  —  (—  2X2  +  *  +  3)  (**  +  5^  +  3*  —  3) 

^  —  *2  +  3^+3,  mod  7, 
or,  making  use  of  3), 
5*3  —  2x*  +  2X3  —  sx2  +  2x  +  1  —  5  (—  2X2  4-  *  +  3)  (3-**  +  x2  —  sx  —  2) 

=  —  «"  +  3*  +  3,  mod  7 ; 
that  is, 

j^— a**4-2jr*-f-  5«*  +  ar+  i—  (—  3X2  —  2x  +  1)  (3^  +  ^  —  5*  — 2) 

=  —  x2  +  3x  +  3,  mod  7, 

where  — 3X2  —  2X  +  1   and  — .r2 -f  3*  +  3  are   the   required  polynomials 
Q(x)  and  R(x). 

§  19.  Congruence  of  two  Polynomials  with  Respect  to  a 
Double  Modulus. 

Two  polynomials,  f±{x),  f2(x).  are  said  to  be  identically  con- 
gruent to  each  other  with  respect  to  the  double  modulus  p,  <f>(x), 
where  p  is  a  prime  number  and  <f>(x)  a  polynomial,  if  their  differ- 
ence, fx{x) — f2{x),  is  divisible,  mod  p,  by  <f>(x);  that  is,  in 
symbols 

/*(*)■»/,(*),  modd/>,  4>(x),  1) 

if  A(*)—  /,(*)  £§(?(*)*(*),  modp,  2) 

or,  in  other  words,  if 

AC*)—  U(x)=*Q{x)*{x)+F{xy.p,  3) 

where  Q(x)  and  F(x)  are  polynomials. 

It  should  be  observed  that  1),  2)  and  3)  all  express  exactly  the 
same  relation  between  the  polynomials  f1(x),  f2{x)  and  <f>(x) 
and  the  prime  number  p,  but,  just  as  in  the  case  of  congruences 
between  integers,  1)  places  this  relation  before  us  in  a  more 
illuminating  manner  than  does  either  2)  or  3). 
6 


82  THE   RATIONAL   REALM CONGRUENCES. 

The  fact  that  f(x)  is  divisible,  mod  p,  by  <f>(x)  is  expressed  in 
the  above  notation  by  writing 

/(*)=o,  modd/>,  <j>(x). 
Ex.     From  §  15,  Ex.,  we  have 

*8  +  3*4  —  4-^  +  2  =  0,  modd5,  2X2  —  3. 
We  have  as  consequences  of  the  above  definition  just  as  in  the 
case  of  integers,  the  double  modulus  p,  <f>(x)  being  understood, 
throughout. 

h(i)mf,(s) 

f*(*)mft{*)> 
ft(x)  ■/,(*). 

fl(x)mft(x) 

U(s)mf£x) 

momial, 

F(x)f1(x)^F(x)f2(x). 

fx{x)mft(x) 

F1(x)=F2(x), 

Ux)-F^x)^ft{x)-Ft{x), 

ft(x)mft(x), 

The  results  corresponding  to  v,  •  •  •,  ix,  §  1,  follow  easily. 

§  20.  Unique  Factorization  Theorem  for  Polynomials  with 
respect  to  a  Prime  Modulus. 

We  shall  now  show  that  a  polynomial  can  be  resolved  in  one 
and  but  one  way  with  respect  to  a  prime  modulus,  p,  into  prime 
factors,  considering  always  associated  factors  as  the  same.  The 
proof  will  be  closely  analogous  to  that  of  the  corresponding 


l  If 

and 

then 

ii.  if 

and 

then 

iii.  // 

and  F(x) 

be  any 

then 

iv.  // 

and 

then 

and,  in  particular, 

if 

then 

THE   RATIONAL   REALM CONGRUENCES.  83 

theorem  for  rational  integers.  We  begin  by  stating  the  following 
theorem  which  is  an  immediate  consequence  of  the  definition  of 
divisibility. 

Theorem  14.  If  f{x)  =  Q(x)<f>(jy)  -f  R(x),  mod  p,  every 
polynomial  that  divides,  mod  p,  both  f(x)  and  <f>(x)  divides  both 
<f>(x)  and  R(x),  and  vice  versa;  that  is,  the  common  divisors, 
mod  p,  of  f(x)  and  <f>(x)  are  identical  with  the  common  divisors, 
mod  p,  of  <f>(x)  and  R(x). 

By  means  of  this  theorem  and  Th.  13  we  can  now  prove  the 
theorem  which  is  the  basis  of  the  unique  factorization  theorem. 

Theorem  15.     //  f1(x),  f2(x)  be  any  two  polynomials  and  p 
a  rational  prime,  there  exists  a  common  divisor,  D(x),  mod  p,  of  . 
fi(*)>  fi(x)  sucn  that  D{x)  is  divisible,  mod  p,  by  every  common  ' 
divisor,  mod  p,  of  f1(x)}  f2(x),  and  there  exist  two  polynomials  ' 
0i(-r)>  <f>2(x)>  sucn  that 

We  may  evidently  assume  f2(x)  of  degree  not  higher  than  fx(x). 
Dividing  f±{x)  by  f2(x),  mod  p,  we  can  find  two  polynomials 
Qx(x),  f3(x),  such  that 

f1(x)  =  Q1(x)f2(x)+fs(x),  modp, 

f&(x)  being  of  lower  degree  than  f2{x). 
Dividing  f2(x)  by  fs(x),  mod  p,  we  have 

f2{x)=Q2(x)fz(x)+fA(x),  modp, 

where  f4(x)  is  of  lower  degree  than  f3(x),  and  similarly 

h{x)  =  Qz{x)f,{x)+U(x),  modp, 


/n_2(*)  ==Qn_2  (>)/«-!  (*)  +/«0),    ™od  p, 

fn.1{x)^Qn.1{x)fn(x),  modp, 
a  chain  of  identical  congruences  in  which  we  must  after  a  finite 
number  of  steps  reach  one  in  which  the  remainder,  fn+1(x),  is  o, 
mod  p,  since  the  degrees  of  the  remainders  continually  decrease. 


84  THE   RATIONAL    REALM CONGRUENCES. 

By  Th.  14  the  common  divisors,  mod  p,  of  fn(x)  and  fn-x(x) 
are  identical  with  those  of  ftut(x)  and  /„_2 (x),  those  of  fn-x(x), 
fn-2(x)  with  those  of  fn-t(x),  fn.s(x),  and  finally  those  of  /3(.r), 
f2(x)  with  those  of  fa(x),  fx(x). 

But  fn(x)  is  a  common  divisor,  mod  p,  of  fn(x)  and  fn-i(x) 
and  is  evidently  divisible  by  every  common  divisor  of  fn(x)  and 
f«-i(x).  Hence  fn(x)  is  the  desired  common  divisor  D(x),  mod 
/>,  of  fx(x)  and/2(». 

If  now  we  substitute  the  value  of  fs(x)  in  terms  of  f±(x), 
f2(x)  obtained  from  the  first  of  these  congruences  in  the  second 
and  the  values  of  /3(.r)  and  f4(x)  in  terms  of' fx(x) ,  f2(x)  in  the 
third  and  continue  until  the  congruence 

/n_2(*)  =Q„_2 (*)/«-i (-0  +/nO),  modp, 

is  reached,  we  shall  obtain  the  congruence 

fx(x)4>t(x)  +ft(x)4>t(x)mD(x)s  modp. 

Cor.  If  fx(x),  f2(x)  be  two  polynomials  prime  to  each  other, 
mod  p,  there  exist  two  polynomials  <f>x(x),  <f>2(x)  such  that 

AOO^O)  +  /2(.r)*2(*)  =  i,  modp. 

In  this  case  D(x)  is  an  integer  a  not  divisible  by  p,  and  we 
have  two  polynomials  ^(x),  $2(-r)  such  that 

ft(x)^(x)  +ft(x)*2(x)w&a,  modp, 

whence,  multiplying  by  the  reciprocal  of  a,  mod  p,  we  obtain 

/i(-0<£iO)  +  /*(*)**(*) ■■  h  modp. 
It  will  be  noticed  that  this  corollary  corresponds  to  Th.  B, 
while  Th.  15  corresponds  to  the  corollary  to  Th.  B,  the  order  of 
proof  here  being  reversed.     The  corollary  could  have  been  proved 
first  as  before.1 

Theorem  16.  //  the  product  of  two  polynomials,  fx{x) ,  f2(x) , 
be  divisible,  mod  p,  by  a  prime  polynomial,  P(x),  at  least  one  of 
the  polynomials,  fx{x),  f2{x),  is  divisible,  mod  p,  by  P(x). 

Let  f1(x)f2(x)  =  Q(x)P(x),modp,  1) 

1  Laurent :    Theorie  des  Nombres  Ordinaires  et  Algebriques,  p.  120. 


THE    RATIONAL    REALM CONGRUENCES.  85 

where  Q(x)  is  a  polynomial,  and  assume  fx(x)  not  divisible,  mod 
p,  by  P{x).  Then  fx{x)  and  P(x)  are  prime,  mod  p,  to  each 
other  and  by  the  last  theorem  there  exist  two  polynomials,  <\>1{x)1 
<f>2(x),  such  that 

£(*)*>(*) +P(*)^(*)  « I,  mod  p.  2) 

Multiplying  2)  by  f2(x),  we  have 

&(*)/,(*)*{*)  +/2(-r)P(.r)^(.r)=/2(^),  mod/», 

and  therefore,  making  use  of  1), 

P(.r)(Q(.r)^(.r)  +  /2(jt)^(jt) )  ==f2(x),  modp, 

where  Q(x)<f>1(x)  -j-f2(x)<f>2(x)  is  a  polynomial.  Hence  /2(-r) 
is  divisible,  mod  p,  by  P(x).  Expressed  in  the  double  modulus 
notation  this  theorem  is: 

If  fi(x)>  /*(«*)  be  any  two  polynomials  and  P{x)  a  prime  poly- 
nomial, mod  p,  and  if 

h(x)f2(x)=o,  modd/>,  P(x), 

then  either  f1(x)^o,  modd/>,  P(x), 

or  /,(.r)=o,  modd/>,  P(x). 

Cor.  1.  If  the  product  of  any  number  of  polynomials  be  divis- 
ible,  mod  p,  by  a  prime  polynomial  P(x),  then  at  least  one  of  the 
polynomials  is  divisible,  mod  p,  by  P(x). 

Cor.  2.  If  neither  of  two  polynomials  be  divisible,  mod  p,  by  a 
prime  polynomial  P(x)%  their  product  is  not  divisible,  mod  p, 
by  P(x). 

Theorem  17.  A  polynomial,  f(x),  can  be  resolved  in  one  and 
but  one  way  into  a  product  of  prime  polynomials,  mod  p. 

Let  f(x)  be  any  polynomial.  We  shall  take  f(x)  in  its  reduced 
form,  mod  p,  for  the  sake  of  convenience,  this  assumption  in  no 
wise  limiting  the  generality  of  the  proof.  Let  the  degree,  mod  p, 
of  f(x)  be  n.  If  f(x)  be  prime,  mod  p,  the  theorem  is  evident. 
If  f(x)  be  not  prime,  it  has  a  divisor,  <f>(x),  mod  p,  and  we  have 

f(x)=<j>{x)*{x),  modp, 


86  THE   RATIONAL   REALM CONGRUENCES. 

where  <f>(x),  &(x)  are  polynomials  neither  of  which  is  a  unit  and 
the  sum  of  whose  degrees  is  n. 

If  <f>(x)  be  not  a  prime  polynomial,  mod  p,  then 

,  +(jc)Wk^i*y*x{x)%  modp, 

where  ^(x),  &i(x)  are  polynomials  that  are  not  units  and  that 
have  degrees  whose  sum  is  equal  to  the  degree  of  <f>(x). 

If  <f>2(x)  be  not  a  prime  polynomial,  mod  p,  we  proceed  in  the 
same  manner  and,  since  the  degrees  of  the  factors  form  a  decreas- 
ing series  of  positive  rational  integers,  we  must  after  a  finite 
number  of  such  factorizations  reach  in  the  series  4>(x),  <t>i(x), 
<f>2(x),  •  •  •  a  prime  polynomial  Px(x),  mod  p.     We  have  then 

f(x)=P1(x)f1(x),  modp. 

Proceeding  similarly  with  fx(x)  in  case  it  be  not  prime,  mod  p, 
we  obtain 

U{x)wiP*(j*)f%{s)t  modp, 
where  P2(x)  is  prime,  mod  p,  and  hence 

f(x)=P1(x)P2(x)f2(x),  modp. 

Continuing  this  process,  we  must  after  a  finite  number  of  such 
factorizations  reach  in  the  series  f(x),  ft(x),  f2(x),--  a  prime 
polynomial  Pn(x),  mod  p.     We  have  then 

f(x)=P1(x)P2(x)  --'Pn(x),  modp, 

where P1(x),P2(x),  •-■,Pn(x)  are  all  prime,  mod  p;  that  is,  f(x) 
can  be  resolved,  mod  p,  into  a  finite  number  of  prime  factors. 

It  remains  to  be  shown  that  this  resolution  is  unique.  Suppose 
that 

f{symQi(s)Q9(s)  "-Qm(x),  modp, 

be  a  second  resolution  of  f(x)  into  prime  factors,  mod  p.     Then 

P1(x)P2(x)  -Pn(x)=Q1(x)Q2(x)  -Qm(x),  modp,      3) 

and  it  follows  from  Th.  16,  Cor.  1  that  at  least  one  of  the  Q(jtr)'s, 
say  Qx(x),  is  divisible,  mod  p,  by  P1(x)  and  hence  is  associated, 
mod  p,  with  Px(x)  ;  that  is, 

Q1(x)^a1P1(x),  modp, 

where  ax  is  a  unit,  mod  p. 


THE   RATIONAL   REALM — CONGRUENCES.  87 

Dividing  3)  by  Px(x),  mod  p,  we  have 

P2(x)  -'-Pn(x)^aiQ2(x)  "-Qn(x),  modp.  4) 

From  4)  it  follows  that  at  least  one  of  the  remaining  Q(x)'s 
must  be  associated,  mod  p,  with  P2{x).  Dividing  4)  by  P2(x), 
mod  p,  and  proceeding  as  before,  we  see  that  with  each  P{x) 
there  is  associated,  mod  p,  at  least  one  Q  (x)  and,  if  two  or  more 
P(x)'s  are  associated,  mod  p,  with  one  another,  at  least  as  many 
Q(x)'s  are  associated,  mod  p,  with  these  P(x)'s  and  hence  with 
one  another. 

In  exactly  the  same  manner,  we  can  prove  that  with  each  Q(x) 
there  is  associated,  mod  p,  at  least  one  P{x)  and,  if  two  or  more 
Q(x)'s  are  associated,  mod  p,  with  one  another,  at  least  as  many 
P(x)'s  are  associated,  mod  p,  with  these  Q(x)'s  and  hence  with 
one  another. 

Hence,  considering  two  associated  factors  as  the  same,  the 
resolutions  are  identical;  that  is,  if  in  the  one  resolution  there 
occur  e  factors  associated,  mod  p,  with  a  certain  prime  polynomial, 
there  will  be  in  the  other  resolution  exactly  e  factors  associated, 
mod  p,  with  the  same  prime  polynomial. 

We  can  now  evidently  write  any  polynomial,  f(x),  in  the  form 

/(#)«(*»<*))*'(/»,(*))•...  (P.Cjt))-,  mod ^ 

where  P1(x),P2(x),  •••,P„(or)  are  the  unassociated  prime  fac- 
tors, mod  p,  of  f(x). 

If  we  take  P1(x),P2(x),  ••',Pn{x)  primary,  the  resolution  is 
absolutely  unique.  The  representations  of  the  greatest  common 
divisor  and  least  common  multiple  given  for  rational  integers  are 
easily  extended  to  polynomials. 

§  21.  Resolution  of  a  Polynomial  into  its  Prime  Factors  with 
respect  to  a  Prime  Modulus. 

The  resolution  of  a  polynomial,  f(x),  into  its  prime  factors, 
mod  p,  may  be  effected  by  dividing,  mod  p,  f(x)  by  each  of  the 
prime  polynomials  of  the  first  degree  x,x — i,---,x  —  p -\- i,(p 
being  taken  positive)  in  turn  until  either  a  polynomial  is  found 
that  divides  f(x),  or  it  is  determined  that  f(x)  is  divisible  by 
none  of  them. 


88  THE   RATIONAL   REALM CONGRUENCES. 

Suppose  that  f(x)  is  divisible,  mod  p,  by  x —  ax  and  that  the 
quotient  is  fx(x).  We  proceed  in  the  same  way  with  fx(x)  until 
we  have  found  all  the  prime,  mod  p,  factors  of  the  first  degree 
oif(x). 

Suppose  that 

f(x)  =  (x  —  a1)(x  —  a2)  •••  (x  —  a„)f2(x),  modp, 

where  f2(x)   has  no   factor,  mod  p,  of  degree  lower  than  the 
second. 

The  prime  factors,  mod  p,  of  the  second  degree  of  f2(x)  can 
next  be  determined  in  the  same  manner,  then  those  of  the  third 
degree,  etc.  In  case,  however,  we  do  not  know  the  prime,  mod  p, 
polynomials  of  the  second  degree,  we  can  simply  determine 
whether  f2(x)  is  divisible,  mod  p,  by  any  polynomial  of  the  second 
degree.  If  it  is,  such  a  polynomial  is  evidently  a  prime,  mod  p, 
polynomial,  for  f2(x)  contains  no  factors,  mod  p,  of  degree  lower 
than  the  second.  The  same  method  can  be  applied  to  the  deter- 
mination of  the  prime  factors  of  higher  degree. 

§  12.  The  General  Congruence  of  the  nth  Degree  in  one 
Unknown  and  with  Prime  Modulus. 

Theorem  18.     //  r  be  a  root  of  the  congruence 

f(x)  —  a0xn  -f  OjX*-1  +  ■••  -\-an  =  o,  mod p,  i ) 

f(x)  is  divisible,  mod  p,  by  x  —  r,  and  conversely,  if  f{x)  be  divis- 
ible, mod  p,  by  x  —  r,  r  is  a  root  of  i). 
Dividing,  mod  p,  f(x)  by  x  —  r,  we  have 

f(x)m{x-~ r)*(*)+*(r),  mod/>, 

whence,  since  r  is  a  root  of  i), 

R(r)  =o,  modp, 

and  hence  f(x)  =  (x  —  r)<f>(x),  modp; 

that  is,  f(x)   is  divisible,  mod  p,  by  x — r.     The   converse   is 
evident. 

If  f(x)  be  prime,  mod  p,  the  congruence  i)  evidently  has  no 
roots.     The  converse  is,  however,  not  true;  that  is,  f(x)  may  be 

1  Borel  et  Drach :    p.  36. 


THE   RATIONAL   REALM CONGRUENCES.  89 

composite,  mod  p,  but  i)  have  no  roots,  for  the  prime,  mod  p, 
factors  of  f(x)  may  all  be  of  higher  degree  than  the  first.  This 
theorem  gives  us  another  method  for  determining  the  factors, 
mod  p,  of  the  first  degree  of  any  polynomial  in  x.  Some  of  these 
factors  may  be  alike  and  we  are  led  therefore  to  say  that  r  is  a 
multiple  root  of  order  e  of  i),  if  f{x)  be  divisible,  mod  p,  by 
(x  —  r)e,  but  not  by  (x  —  r)e+1. 

If  therefore  rlfr2i'",rm  be  the  incongruent  roots  of  i)  of 
orders  et,e2,---,em  respectively,  we  have 

f(x)  =  (x  —  rx)*(x~  r2)e*...  (x  —  rm)e'-f1(x),  modp, 

where  f1(x)  is  a  polynomial  having  no  linear  factor,  mod  p,  and 
whose  degree,  s,  is  such  that 

^1  +  ^2  +  "•  +em  +  s=n, 
where  n  is  the  degree  of  f(x). 

Counting  a  multiple  root  of  order  e  of  i)  as  e  roots,  we  see  that 

1)  has  exactly  as  many  roots  as  f(x)  has  linear  factors,  mod  p, 
and  obtain  the  following  important  theorem: 

Theorem  19.     The  number  of  roots  of  the  congruence 

f(x)  =  a0xn  +  axX**  *\ +  0»seO,  mod  p, 

where  p  is  a  prime  number,  is  not  greater  than  its  degree. 

Cor.  1.  If  the  number  of  incongruent  roots  of  a  congruence 
with  prime  modulus  be  greater  than  its  degree  the  congruence  is 
an  identical  one. 

Cor.  2.     If  the  congruence 

f(x)  =0,  mod  p,  2) 

have  exactly  as  many  roots  as  its  degree  and  cf>(x)  be  a  divisor, 
mod  p,  of  f(x),  then  the  congruence 

(f>(x)  =0,  mod p, 

has  exactly  as  many  roots  as  its  degree;  for 

f(x)==<l>(x)Q(x),  modp, 

where  Q(x)  is  a  polynomial  in  x,  and  every  root  of  the  congruence 

2)  is  a  root  of  either  the  congruence 

<£(»===  o,  modp,  3) 


gO  THE   RATIONAL   REALM — CONGRUENCES. 

or  of  the  congruence 

Q(jr)==o,  mod  p.  4) 

Moreover,  the  sum  of  the  degrees  of  3)  and  4)  is  equal  to  the 
degree  of  2).  If,  therefore,  <f>(x)  had  fewer  roots  than  its  degree, 
then  Q  (x)  must  have  more  roots  than  its  degree,  which  is  impos- 
sible.    Hence  the  corollary. 

§  23.  The  Congruence  x0(m)  —  1  =  0,  mod  nu 
Although  in  the  case  of  congruences  of  degree  higher  than  the 
first  the  theorem  just  given  tells  all  that  we  know  in  general 
regarding  the  number  of  their  roots,  still  there  is  one  important 
case  in  which  the  number  of  roots  is  always  exactly  equal  to  the 
degree  of  the  congruence. 

Theorem  20.     The  congruence 

x<p(m->  —  i===o,  modw,  1) 

has  exactly  as  many  roots  as  its  degree. 

The  <]>(nt)  integers  of  a  reduced  residue  system,  mod  m,  evi- 
dently satisfy  1).  Moreover,  since  by  §1,  ix,  two  integers  con- 
gruent, mod  m,  have  with  m  the  same  greatest  common  divisor, 
and  the  greatest  common  divisor  of  1  and  m  is  1,  every  root  of  1) 
must  have  with  m  the  greatest  common  divisor  1,  that  is,  be  prime 
to  m.  Hence  the  number  of  roots  of  1)  is  exactly  equal  to  <£(m), 
its  degree. 

Ex.    The  congruence 

#*C10)  —  1  ==0,  mod  10, 
or  x*  —  1^0,  mod  10, 

has  the  four  roots  1,  3,  7,  and  9. 

Cor.    If  d  be  a  positive  divisor  of  p  —  1,  the  congruence 

xd  —  1  =  o,  mod  p, 

where  p  is  a  prime,  has  exactly  d  roots;  for  xd  —  /  is  a  divisor 
of  x*-x  —  1  and  hence  by  Th.  19,  Cor.  2,  we  have  the  corollary. 
Since  the  congruence 

xp  —  x===o,  modp, 


THE  RATIONAL   REALM — CONGRUENCES.  9 1 

has  the  p  roots  o,  i,  2,  •  •  -,  p  —  1  equal  in  number  to  its  degree,  we 
have  the  identical  congruence 


xp  —  x  =  x(x — %)(x  —  2)  •••  (x —  p — 1),  mod  p. 

Ex.     X1  —  x  =  x(x  —  i)0  —  2)0  —  3)0  —  4)0  —  5)0  —  6),  mod  7. 
§  24.    Wilson's  Theorem. 

The  result  just  obtained  gives  us  a  proof  of  the  following  inter- 
esting theorem. 

Theorem  21.  If  p  be  a  prime  number  and  rx,r2,  •••,r$(p)  be  a 
reduced  residue  system,  mod  p,  then 

Vi  •  •  •  Uip)  +  1=0,  mod  p. 
By  the  previous  section  we  have  evidently 

x<t>^  —  i  =  {x  —  r1){x  —  r2)  •••  (*  —  !>(„,),  modp, 
from  which,  putting  x  =  o,  we  have 

—  i  =  (—  r1)(—  r2)  •••  (—  r+m}t  modp, 
whence,  since  <f>(p)  is  even  except  when  p  =  2, 
Vj  "-^(p))  +  1  =  0,  modp, 

which  evidently  holds  also  when  p  =  2.1 

Ex.     Let  p  =  5,  and  take  as  a  reduced  residue  system,  mod  5,  the  integers 

—  2,  —  1,  1,  2.    Then 

(—  2)(—  i).i.2  +  i  =  5  =  o,  mod  5. 

This  theorem  is  a  particular  case  of  the  following  more  general 
theorem  that  is  due  to  Gauss.2 

If  ri>r2>'">r<i>(m)  be  a  reduced  residue  system,  mod  m,  the 

product  rxr2-~  r^(m)  is  congruent  to  — i,  mod  m,  when  m  =  4, 

pn  or  2pn,  where  p  is  an  odd  prime,  and  is  congruent  to  1,  mod  m, 

when  m  has  any  other  value. 

The  two  following  examples  will  illustrate  this  theorem;  for 
its  proof  see  references  given  above. 

Ex.  1.     Let  m  =  f,  and  take  as  a  reduced  residue  system,  mod32,  — 4, 

—  2,  — 1,  1,  2,  4;    then 

(—  4)(—  2)(—  1). 1-2-4  =  —  64  =  —  1,  mod32. 

1  See  Matthews,  §  16,  for  another  proof  of  this  theorem. 

2  Gauss:  Disq.  Arith.,  Art.  78.  Dirichlet-Dedekind :  §38.  Bachmann: 
Niedere  Zahlentheorie,  p.  170.     Cahen :    p.  103. 


92  THE   RATIONAL   REALM CONGRUENCES. 

Ex.  2.  Let  m  =  15,  and  take  as  a  reduced  residue  system,  mod  15,  — 7, 
—  4,-2,-1,  1,  2,  4,  7;   then 

(—7)  (—4)  (—2)  (—  1)  -1-2.4.7  =  3136=  1,  mod  15. 

As  a  special  case  of  Th.  21  we  have  the  following: 
If  p  be  a  positive  prime  number  and  the  product  of  all  positive 
integers  less  than  p  be  increased  by  1,  the  result  is  divisible  by  p; 
that  is, 

(p  —  1 )  !  +  1  sb  o,  mod  p. 
The  theorem  was  first  stated  in  this  form  by  Waring  in  his  "  Medi- 
tationes  Algebraicae  "  (1770)  and  ascribed  to  its  author,  Sir  John 
Wilson. 

The  converse  of  the  original  form  is  true ;  that  is,  //  the  product 
of  all  positive  integers  less  than  a  given  integer,  m,  be  increased 
by  1  and  the  result  be  divisible  by  m,  then  m  is  a  prime  number. 
This  is  easily  seen  to  be  true;  for,  if  m  =  ab,  where  neither  a  nor 
b  is  a  unit,  then  (m  —  1)  !  is  divisible  by  a,  whence  we  have 

(m  —  1 )  !  +  I  W^  o,  mod  m. 

For  example  5  !  +  1  =  121  ^o,  mod  6. 

Wilson's  theorem  gives  therefore  an  unfailing  method  for  deter- 
mining whether  any  given  integer  is  a  prime  number.  It  is,  how- 
ever, obviously  of  no  practical  use  on  account  of  the  immense 
labor  of  the  numerical  reckoning  when  m  is  large. 

§  25.     Common  Roots  of  Two  Congruences. 
The  common  roots  of  two  congruences 

f1(x)=o,  modp,  and  /2(jr)=o,  modp, 

are  evidently  the  roots  of  the  congruence 

<f>(x)  =0,  modp, 

where  4>(x)  ls  tne  greatest  common  divisor,  mod  p,  of  fx{x)  and 
/2(.r).     Since  the  congruence 

xp  —  ;r  =  o,  modp,  1) 

has  for  its  roots  the  numbers  of  a  complete  residue  system,  mod 
p,  the  incongruent  roots  of  any  congruence 

/(,r)  =0,  modp, 


THE   RATIONAL   REALM CONGRUENCES.  93 

will  be  the  roots  of  the  congruence 

<f>(x)  =0,  modp,  2) 

where  cf>(x)  is  the  greatest  common  divisor,  mod  p,  of  x? —  x 
and  f(x).  This  gives  us  another  method  of  determining  all  the 
incongruent  roots  of  any  given  congruence  with  prime  modulus. 
The  congruence  2)  will  always  have  as  many  roots  as  its  degree, 
since  the  congruence  1)  has  as  many  roots  as  its  degree  and  <j>(x) 
is  a  divisor,  mod  p,  of  x?  —  x. 

Ex.     To  find  the   roots   of  the  congruence 

xl  —  sx3 —  x2-\-2x  —  6  =  0,  mod  7,  3) 

by  the  above  method,  since  o  is  not  a  root  of  the  congruence,  we  need 
only  find  the  greatest  common  divisor,  mod  7,  of  xl  —  3x*  —  x2-\-2x —  6 
and  xe—  1. 

This  -greatest  common  divisor  is  x2  —  3X  -\-  2,  and  the  congruence 

x2  —  3^  +  2^0,  mod  7, 
has  the  roots  1  and  2,  that  are  therefore  the  incongruent  roots  of  3). 

§  26.    Determination  of  the  Multiple  Roots  of  a  Congruence 
with  Prime  Modulus. 

The  multiple  roots  of  the  congruence 

f(x)  =0,  modp,  1) 

may  be  determined  by  a  method  exactly  analogous  to  that  em- 
ployed for  determining  the  multiple  roots  of  an  algebraic  equation. 
Thus  let  P(x)  be  a  prime  function,  mod  p,  and  let  f(x)  be  divis- 
ible, mod  p,  by  (P(x))e  but  not  by  (P(x))e+1;  then 

f(x)==(P{x)YQ{x),  modp, 

or,  what  is  the  same  thing, 

f{x)  =  {P(x)YQ{x)+pF(x),  2) 

where  F(x)  and  Q(x)  are  polynomials  in  x  and  Q(x)  is  prime, 
mod  p,  to  P{x). 

Differentiating  2),  we  have 

f(x)  =  (P(x)y->(eP'(x)Q(x)  +  P(x)Q'{x))  +  pF'(x), 

where  .P'(.r),  Q'(x')  and  F'(x)  are  polynomials  in  x.     Hence 

f(x)  =  (P(x))^Q1(x),  modp, 


94  THE  RATIONAL   REALM CONGRUENCES. 

where  Qx(x)  is  a  polynomial  in  x  and  is  moreover  not  divisible, 
mod  p,  by  P(x),  for 

Qx(x)=eP'(x)Q(x)  +P(x)Q'(x), 

where  P'{x)  is  of  lower  degree  than  P(x)  and  Q(x)  is  prime, 
mod  p,  to  P(x).  Therefore  f(x)  is  divisible,  mod  p,  by  the 
prime  factor  P(x)  exactly  once  less  than  f(x)  is  divisible  by 
P(x).  In  particular,  if  f(x)  be  divisible,  mod  p,  by  {x  —  r)e, 
but  not  by  (x  —  r)e+1,  then  f(x)  is  divisible,  modp,  by  (# —  r)e_1 
but  not  by  (x  —  r)e.     Hence  the  theorem: 

Theorem  22.    //  the  congruence 

f(x)  =  0,  modp, 

have  a  multiple  root  r  of  order  e,  the  congruence 

f(x)  =  0,  modp, 

has  the  multiple  root  r  of  order  e  —  1. 

If  the  greatest  common  divisor,  mod  p,  of  f(x)  and  f(x)  be 
<f>(x),  then  the  roots  of  the  congruence 

4>0)=o,  modp,  3) 

if  it  have  any,  will  be  the  multiple  roots  of  1)  and  each  root  of 
3)  will  occur  once  oftener  as  a  root  of  1)  than  as  a  root  of  3). 

It  may  happen,  of  course,  that  f(x)  and  f(x)  have  a  common 
divisor,  <f>(x),  mod  p,  and  yet  1)  has  no  multiple  roots.  In  this 
case  the  repeated  prime  factors,  mod  p,  of  f(x)  are  of  higher 
degree  than  the  first,  and  <f>(x)  therefore  contains  no  factor  of 
the  first  degree,  mod  p. 

Ex.    Let  the  given  congruence  be 

f(x)=2x8  —  * -|- 1  ^  °>  mod  5-  4) 

We  have  f(x)=6x2 — i^^r — 1,  mod  5, 

and  the  greatest  common  divisor,  mods,  of  /(■*")  and  f(x)  is  x -\- 1. 

The  congruence 

x  +  1  ^  o,  mod  5, 
has  the  root  —  I. 

Hence  the  congruence  4)  has  two  roots  —  1.    Dividing  f(x)  by  (x  +  i)2, 

we  have  f{x)  =2(x -\-  i)2(x  —  2),  mods, 

and  see  that  f(x)  has  the  third  root  2. 


THE  RATIONAL   REALM — CONGRUENCES.  95 

§  2j.  Congruences  in  One  Unknown  and  with  Composite 
Modulus. 

The  solution  of  a  congruence  of  the  form 

f( x)  =  a0xn  +  axxn~x  +  "  +  o»930,  mod  m,  i ) 

where  m  =  mxm2  •  •  ■  mt, 

mlf  m2,  "•  ntt  being  integers  prime  each  to  each,  can  be  reduced  to 
the  solution  of  the  system  of  t  congruences, 

f(x)  s=o,  modMj,! 
f(x)  =o,  mod  w2, 

;  :      r  2) 

f(x)z==o,  modwf.J 

Every  root  of  i)  is  evidently  a  root  of  each  of  the  congruences 
2),  and  conversely  any  integer,  that  is  simultaneously  a  root  of 
each  of  the  congruences  2),  is  a  root  of«i). 

If  therefore  alfa2,  --,at  be  roots  of  the  congruences  2)  and  r 

be  chosen  so  that 

r  =  a19  mod  mlt  ' 
r==a2,  modm2, 

I    :      :       l  3) 

rz=at,  mod  mt,  - 
then  r  is  a  root  of  1). 

Since  mlf  m2,  •  •  •,  mt  are  prime  each  to  each,  it  is,  by  §14,  always 
possible  to  find  r  so  as  to  satisfy  the  conditions  3). 

Let  blt  b2,---,bt  be  auxiliary  integers  selected  as  in  §  14;  then 

r  =  a1b1  +  a2b2  +  •  •  •  +  atbt,  mod  m  4) 

is  a  root  of  1),  and,  if  the  congruences  2)  have  respectively 
hyh>'"Jt  incongruent  roots,  then  by  §14  1)  has  lxl2--lt  incon- 
gruent  roots,  that  are  obtained  by  putting  in  4)  for  ar,a2i--,at 
respectively  the  l^,l2,-",h  roots  of  the  congruences  2). 

In  particular,  if  any  one  of  the  congruences  2)  have  no  root, 
then  1)  has  no  root. 

Ex.    The  solution  of  the  congruence 

**  +  3*3  +  3**  +  3*  +  2  =  0,  mod  30,  5) 


96  'THE   RATIONAL   REALM CONGRUENCES. 

can  be  reduced  to  the  solution  of  the  two  congruences 

x*  +  3X3  -f  sx-  -+-  $x  -\-  2  ==  o,  mod   6,  6) 

and  x*  +  3x3  +  sx2  +  3x  +  2  =  o,  mod   5.  7) 

The  roots  of  6)  are  — 2,  —  1,  1,  2  and  those  of  7)  are  — 2,  —2,  — 1,  2. 
The  roots  of  5)  are  then 

r.Oj  = — 2,  —  I,  I,  a.1 

r==  25* +  6a2,  mod  30.  j  a__2__^  2 

that  gives  as  the  roots  of  5),  —  13,  — II,  — 8,  — 7,  — 2,  —  1,2,4,7,8, 13,  14. 
If  now  we  suppose  m  to  be  resolved  into  a  product  of  powers 
of  its  different  prime  factors,  that  is, 

m=p1eip2e2  •••  prer, 

where  pltp2,  "',pr  are  different  primes,  then  the  solution  of  1)  is 
reduced  to  the  solution  of  n  congruences  of  the  form 

/(.r)=o,  modpe.  8) 

We  shall  now  show  tfiat  the  solution  of  8)  can  be  made  to 
depend  upon  the  solution  of  the  congruence 

/0)=o,  mod^"1,  9) 

where  the  modulus  is  a  power  of  p  one  degree  lower  than  that  of 
the  modulus  of  8),  and  thus  be  made  to  depend  eventually  upon 
the  solution  of  the  congruence 

f(x)  =0,  modp, 

whose  modulus  is  a  prime. 

Let  x9  be  a  root  of  9)  ;  then  all  integers  of  the  form  x0  +  pe~1y, 
where  y  is  an  integer,  are  roots  of  9).  Furthermore,  since  all 
roots  of  8)  are  roots  of  9),  if  8)  have  roots  they  must  be  of  this 
form. 

Putting  in  8)  x  =  x0  +  pe~xy,  10) 

we  have  f(x0  -j-  pe~xy)  ^  o,  mod  pe, 

or,  expanding  f(x0  +  p^y),  / 

/K)  +f(^)Pe-1y+f-^P2e-'Y-  +  -^f  mod J-.      11) 

Since  f(*0)  =0,  mod/>e_1, 

1  See  Example  §  14. 


THE   RATIONAL   REALM — CONGRUENCES.  97 

we  have  f(*o)=cpe~1, 

and  hence,  dividing  each  term  of  n)  by  />e_1, 

c  +  fWj+^f¥  +  "^o,  mod?, 

whence  we  have 

c  -\-f(x0)y  =  o,  modp,  12) 

as  a  necessary  and  sufficient  condition  that  y  must  satisfy  in  order 
that  the  root,  x0  +  pe~xy,  of  9)  may  also  be  a  root  of  8). 
There  are  three  cases  to  be  considered: 

i.  If  f(x9)qk6s  modp, 

there  is  always  one  and  but  one  value,  y0,  of  y  that  satisfies  12) 
and  this  gives  one  value  only  of  x0  +  pe'xy^  tnat  satisfies  8). 

ii.  If  f(x0)==o,  modp,  and  c^£o,  modp, 

there  is  no  value  of  y  satisfying  12)  and  hence  no  value  of  x  of 
the  form  x0  +  pe'1y  satisfying  8)  ;  that  is,  8)  has  no  root. 

iii.  If  f(x0)^=o,  modp,  and  c  =  o,  modp, 

then  12)  is  an  identical  congruence  and  consequently  12)  has  \p\ 
solutions,  mod  p,  from  which  by  substitution  in  10)  we  obtain  \p\ 
solutions  of  8).1 

Ex.    The  roots  of  the  congruence 

x*  —  8x*  +  9**  +  9*  +14  =  ©,  mod  52,  13) 

if  any  exist,  must  satisfy  the  congruence 

x*  —  8-r3  +  9x~  +  9*  +  14  aa  0,  -mod  5, 
whose  roots  are  1  and  2,  and  hence  be  of  the  form 
i+5y    or    2-j-Sy. 
Substituting  1  +  5y  and  2  -f-  5y  in  13),  we  obtain  respectively 

5+   7y  =  o,  mod  5,  14) 

and  4 — io:y  =  o,  mods.  15) 

From  14)  we  have  y  as  o,  mod  5, 

and  from  15)  ;y==i,  mods, 

that  give  1  and  7  as  the  roots  of  13). 
1  See  Cahen  :•   pp.  96-103. 
7 


98  THE   RATIONAL   REALM CONGRUENCES. 

§28.    Residues  of  Powers. 

//  a  be  prime  to  m,  and  b^aff  mod  m,  where  t  is  a  positive 
integer,  b  is  said  to  be  a  power  residue  of  a  with  respect  to  the. 
modulus  m. 

For  example,  since  4==32,  mod  5,  we  say  that  4  is  a  power  resi- 
due of  3  with  respect  to  the  modulus  5. 

Two  power  residues  of  a  which  are  congruent  to  each  other, 
and  hence  to  the  same  power  of  a,  mod  m,  are  looked  upon  as 
the"  same. 

A  system  of  integers  such  that  every  power  residue  of  a,  mod  m, 
is  congruent  to  one  and  only  one  integer  of  the  system,  mod  m,  is 
called  a  complete  system  of  power  residues  of  a  with  respect  to 
the  modulus  m. 

Ex.  Every  power  of  5  is  congruent,  mod  6,  to  1  or  5.  Hence  1,  5 
constitute  a  complete  system  of  power  residues  of  5,  with  respect  to  the 
modulus  6. 

These  integers  may  evidently  be  selected  from  among  the  in- 
tegers of  any  reduced  residue  system,  mod  m.  For  convenience 
they  are  usually  taken  from  the  system  1,2,  •••,  \m\  and  we  may 
indeed  define  a  complete  system  of  power  residues  of  a,  mod  m, 
as  being  the  smallest  positive  residues  that  the  successive  powers 
of  a,  a°=  1,  a1,  a2,  a3,  •••,#',  •■•  give  when  divided  by  m. 

The  more  general  definition  given  above  will,  however,  serve 
our  purposes  better  as  it  will  admit  of  direct  extension  to  realms 
of  higher  degree  than  the  first,  while  the  latter  does  not. 

We  shall  now  investigate  certain  questions  relating  to  power 
residues,  and,  in  particular,  the  important  one  as  to  when  a  com- 
plete system  of  power  residues  of  an  integer  a,  mod  m,  is  also  a 
reduced  residue  system,  mod  m. 

The  following  table  gives  the  power  residues  of  all  numbers  of 
a  reduced  residue  system,  mod  13,  with  respect  to  this  modulus. 
In  order  to  calculate  the  residue  of  ak,  it  is  not  necessary  to  raise 
a  to  the  &th  power,  but  only  to  multiply  the  residue  of  ofc-1  by  a 
and  then  take  the  residue  of  the  product  with  respect  to  m. 

m==  13. 


THE   RATIONAL   REALM CONGRUENCES. 


99 


I 

2 

3 

4 

5 

6 

7 

S 

9 
io 
ii 

12 


I 

4 
9 
3 

12 
IO 
IO 
12 

3 
9 
4 

i 


i 
8 

i 

12 

s 

8 
5 
5 
i 

12 

5 

12 


I 

6 

9 

io 

5 

2 

II 

8 
3 
4 
7 

12 


I 

12 
I 
I 

12 

12 

12 

12 

I 

I 

12 

I 


I 
I  I 

3 

4 

S 

7 
6 

5 

9 
io 

2 
12 


I 

5 

i 

12 

5 

5 

S 

s 
I 

12 


I 

IO 

3 

9 

12 

4 

4 

12 

9 

3 

io 

i 


i 
7 
9 

io 
8 

ii 

2 

5 
3 
4 
6 

12 


We  ask  now,  what  is  the  smallest  value  ta  of  £  other  than  o  for 
which  we  have 

a'=  i,  modw. 

That  ta  always  exists  and  is  ^<j>(m)  is  evident  from  Fermat's 
theorem,  that  gives,  since  a  is  prime  to  m, 
a0(m)==  i}  mod  w. 

Giving  ta  the  above  meaning,  we  say  that  the  integer  a  appertains 
to  the  exponent  ta  with  respect  to  the  modulus  m.  We  see  from 
the  table  that 

2, 6, 7, 1 1  appertain  to  the  exponent  12;  that  is,  ^(13).    " 
4, 10  appertain  to  the  exponent    6 

5,   8  appertain  to  the  exponent   4  r,  mod  13. 

3,   9  appertain  to  the  exponent   3 
12  appertains  to  the  exponent   2 

It  is  evident  that,  if  a=  b,  mod  m,  then  a  and  b  appertain  to  the 
same  exponent,  mod  m. 

Theorem  2^.     If  the  integer  a  appertain  to  the  exponent  ta, 
mod  m,  then  the  ta  powers  of  a, 


1,  a,  a2,  --^V1, 

are  incongruent  each  to  each,  mod  m. 

Let  as  and  as+r  be  any  two  of  the  powers  1).     If 

a8+r==a8,  modm, 

then,  since  a  is  prime  to  m, 

arz=  1,  modw. 


1) 
3) 


100  THE   RATIONAL   REALM CONGRUENCES. 

But  r  <  ta  and  hence  3)  is  impossible,  since  a  appertains  to  ta. 
Therefore  2)  is  impossible. 

Theorem  24.  //  a  appertain  to  the  exponent  ta,  mod  m,  any 
two  powers  of  a  with  positive  exponents  are  congruent  or  incon- 
gruent  to  each  other,  mod  m,  according  as  their  exponents  are 
congruent  or  incongruent,  mod  ta. 

Let  a81,  as*  be  any  two  powers  of  a,  slf  s2  being  positive  integers, 
and  let 

s±  s=a  qxta  +  r1}    s2  =  q2ta  +  r2, 

where  qlf  q2  are  positive  integers  and 

o^rx<ta,     o^r2<ta,     r^r2.  4) 

If  a&**^»a**i+rt,  modwz,  5) 

then  ari==ar2,  mod  m,  6) 

whence,  since  a  is  prime  to  m, 

ar1-r2^Ij  rnodwt. 
But  from  4)  we  have 

o^r1  —  r2<ta, 
and  hence,  since  a  appertains  to  ta,  mod  m, 

r±  =  r2.  7) 

Therefore'  s±  =  s2,  modfa,  8) 

is  a  necessary  condition  for 

asi  =  aS2,  mod  m.  9) 

Moreover,  from  8)  follow  in  turn  7),  6)  and  5). 

Hence  8)  is  also  a  sufficient  condition  for  the  existence  of  9). 
We  have  therefore 


a1  as  ata+1  hbs  a2ta+1  ==  a3ta+1  h=  • 
a2  =  ata+2  sb  a2ta+2  =  a3*a+2  ss  • 

a'""1  =  a2ta~x  be  a3*""1  se  a4*0"1  = 


,  modw. 


This  is  known  as  the  law  of  the  periodicity  of  the  power  resi- 
dues.    It  can  be  verified  by  an  examination  of  the  table,  p.  99, 


51 

= 

5' 

5' 

52 

as 

5C 

510 

53 

= 

57 

S11 

0  =  4  =  8=12  y 
,  mod  13,       *=£_   9  L   mod  4. 


THE   RATIONAL   REALM CONGRUENCES.  IOI 

where  we  see,  for  example,  that  5  appertains  to  the  exponent  4, 
mod  13,  and  we  have 

2  =  6  =  IO 

3  =  7=11  J 

Theorem  25.  The  exponent,  ta,  to  which  an  integer  a  apper- 
tains with  respect  to  the  modulus  m,  is  always  a  divisor  of  <f>{m)} 

Since  a0<w>  ==  1  =  a0,  mod  m, 

we  have  by  Th.  24, 

<f>(m)  =  0,  modta. 

Theorem  26.  If  two  integers,  alf  a2,  appertain,  mod  m,  to  two 
exponents,  tlt  t2,  that  are  prime  to  each  other,  then  their  product, 
axa2,  appertains,  mod  m,  to  the  exponent,  tj2. 

Let  axa2  appertain,  mod  m,  to  an  exponent  t,  then 

(a1a2y*smis  modm.  10) 

Raising  both  members  of  10)  to  the  tx  power,  we  have 
a^a^xt  ==  lf  m0(|  m 

But  a±tlt  aa  1,  mod  m, 

and  hence  a,,*1'—1*  modm, 

and  therefore,  since  a2  appertains  to  the  exponent  t2,  mod  m,  txt 
must  be  a  multiple  of  t2  (Th.  24).  Whence,  since  tx  and  t2  are 
prime  to  each  other,  it  follows  that  t  is  a  multiple  of  t2.  In  like 
manner  we  can  show  that  t  is  a  multiple  of  tx. 

Therefore  t,  being  a  multiple  of  tt  and  t2,  that  are  prime  to  each 
other,  is  a  multiple  of  their  product  txt2.  Hence  the  smallest  pos- 
sible value  of  t  for  which  1)  will  hold  is  tj2,  and  a±a2  appertains 
to  this  exponent,  mod  m. 

Ex.  We  see  from  the  table,  p.  99,  that  12  and  3  appertain,  mod  13,  to 
the  exponents  2  and  3  respectively,  and  that  their  product  36(^  10,  mod  13) 
appertains  to  the  exponent  6. 

Limiting  ourselves  now  to  the  case  in  which  the  modulus  is  a 

1For  a  proof  of  this  theorem  not  dependent  upon  Fermat's  theorem, 
see  Mathews,  p.  18. 


102  THE   RATIONAL   REALM CONGRUENCES. 

prime  number,  p,  we  ask  whether  there  are  integers  appertaining 
to  every  positive  divisor  of  <f>(p)  and,  if  so,  how  many.  Before 
proving  the  theorem,  that  will  answer  this  question  in  its  entirety, 
let  us  examine  the  table,  p.  99,  and  see  how  matters  stand  when 
/>  — 13.  The  positive  divisors  of  ^(13),  =  12,  are  1,  2,  3,  4,  6 
and  12. 

To    1  appertains  the  single  integer    1, 

To   2  appertains  the  single  integer  12, 

To    3  appertain  the  two  integers     3,   9,  I    mo^j^ 

To   4  appertain  the  two  integers     5,    8, 

To    6  appertain  the  two  integers     4, 10, 

To  12  appertain  the  four  integers    2,    6,  7,  11, 

Theorem  27.  To  every  positive  divisor,  t,  of  <t>(p),  there 
appertain  <f>(t)  integers1  with  respect  to  the  modulus  p. 

Assume  that  to  every  positive  divisor,  t,  of  <j>(p),  there  apper- 
tains at  least  one  integer,  a.  We  shall  show  that,  if  this  assump- 
tion be  true,  there  appertain  to  t  <f>(t)  integers;  that  is,  to  every 
positive  divisor,  t,  of  <f>(p)  there  appertain  either  <f>(t)  integers 
or  no  integers.  Let  \J/(t)  denote  the  number  of  integers  apper- 
taining to  t.     Each  of  the  integers 

a°  =  i,a,a2,  -,ow  11) 

is  a  root  of  the  congruence 

x*  =  i,  modp,  12) 

for,  if  ar  be  one  of  these  integers,  then 

(ar)*=(a')r=i,  modp, 

since  a'=i,  modp. 

The  integers  11)  are  moreover  by  Th.  23  incongruent  each 
to  each,  mod  p,  and,  being  t  in  number,  are  therefore  all  the  roots 
of  12),  since  12)  can  not  have  more  than  t  incongruent  roots. 
But  every  integer  appertaining  to  t  must  evidently  be  a  root  of 
12)  and  we  need  look  therefore  only  among  the  integers  11)  to 
find  all  integers  appertaining  to  t.  Let  ar  be  any  one  of  the  in- 
tegers 11).     If  ar  appertain  to  t,  we  must  have  ar,a2r,  •••,a('~1)r 

1  We,  of  course,  consider  only  incongruent  integers ;  see  p.  99. 


THE   RATIONAL   REALM — CONGRUENCES.  IO3 

each  incongruent  to  I,  mod  p.  By  Th.  24  the  necessary  and  suffi- 
cient condition  for  this  is 

ir^to,  modt,  13) 

where  i  runs  through  the  values  1,2,  •••,  t  —  1.  In  order  now  that 
13)  may  hold,  we  must  have  r  prime  to  t ;  for  suppose  that  the 
contrary  is  true  and  that  d  is  the  greatest  common  divisor  of  r 
and  t,  assuming  for  convenience  d  to  be  positive.     We  have 

r=rxd,     t  =  txd, 

and,  since  tx  <  t  and  i  runs  through  all  values  from  1  to  t — 1, 
one  of  the  values  of  t  will  be  tx  and  we  shall  have  for  this  value 

txrxd  =  o>  mod  txd; 

that  is,  13)  does  not  hold. 

But,  since  i  <  t,  13)  holds  whenever  r  is  prime  to  t.  Hence  the 
necessary  and  sufficient  condition  that  any  one,  ar,  of  the  integers 
1,  a,  a2,  •■•,  a*-1  shall  appertain  to  t,  is  that  its  exponent,  r,  shall  be 
prime  to  t.  This  condition  is  fulfilled  by  <f>(t)  of  these  integers, 
and  we  have  proved  therefore  that 

tf,(t)  =  either  </>(/)  oro. 

We  shall  now  prove  that  the  latter  case  can  never  occur.  We 
separate  the  <f>(p)  integers  of  a  reduced  residue  system,  mod  p, 
into  classes  according  to  the  divisor  of  <f>(p)  to  which  they  apper- 
tain; that  is,  if  tlyt2,---,tn  be  the  positive  divisors  of  <f>(p),  we 
put  in  one  class  the  \p{tx)  integers  of  the  above  system  that  apper- 
tain to  tly  in  another  class  the  if/(t2)  integers  that  appertain  to 
£0,  etc.  It  is  evident  that  no  integer  can  belong  to  two  different 
classes  and  that  every  integer  must  belong  to  some  one  of  these 
classes. 

The  integers  of  a  reduced  residue  system,  mod  p,  being  <f>(p) 
in  number,  we  have  therefore 

Hh)  +Hh)  +  -  +tM=<i>(P). 

But  by  Th.  5,  <f>(p)  taking  the  place  of  m,  we  have 
<K'i)  +<t>(t2)  +  •••  +  <f>(tn)  =<£(/>), 


104  THE   RATIONAL   REALM CONGRUENCES. 

whence 

#(«l)  +*(*•)  H hHtn)  =<f>(t1)  +<j>(t2)  +  •  •  •  +<f>(tn).      14) 

Since,  however,  every  term  in  the  first  member  of  14)  is  equal 
either  to  the  corresponding  term  in  the  second  member  or  o,  if 
even  a  single  term  in  the  first  member  were  o,  14)  would  not  hold. 
Hence  no  term  in  the  sum  ^(^)  +  ^(f2)  +""*  +♦(*»)  1S  °- 
Therefore  f(t)  =  <j>(t). 

§  29.    Primitive  Roots. 

An  integer,  that  appertains  to  the  exponent  <f>(m)  with  respect 
to  the  modulus  m,  is  said  to  be  a  primitive  root  of  m. 

For  example;  2,  6,  7  and  11  appertain,  mod  13,  to  the  exponent 
0(13),  =  12,  and  are  therefore  primitive  roots  of  13.  It  can  be 
shown  that  such  integers  exist  only  when  m  =  2,  4,  pn  or  2pn, 
where  p  is  an  odd  prime.1  We  shall  discuss  however  only  the 
case  where  m  is  a  prime  number. 

It  having  been  proved  in  Th.  27  that,  if  p  be  a  prime,  there 
appertain  <j>(<j>(p))  integers  to  the  exponent  <f>(p),  mod  p,  we  see 
that  p  has  always  <f>(<j>(p))  incongruent  primitive  roots.  If  r  be 
a  primitive  root  of  p,  then  by  Th.  23  the  <j>(p)  powers  of  r 
r,r2,---,  r0(p)  form  a  reduced  residue  system,  mod  p.  Hence  every 
integer,  that  is  not  divisible  by  p,  is  congruent  to  one  of  these 
powers  of  r,  mod  p.  This  property,  upon  which  depends  the  use- 
fulness of  a  primitive  root,  may  be  used  to  define  it  as  follows : 
An  integer,  a  complete  system  of  zvhose  pozver  residues,  mod  m, 
constitute  a  reduced  residue  system,  mod  m,  is  called  a  primitive 
root  of  m. 

For  example;  2,  22,  23,  24,  25,  26,  27,  28,  29,  210,  211,  212  con- 
stitute a  reduced  residue  system,  mod  13.  Hence  2  is  a  primitive 
root  of  13. 

We  shall  illustrate  the  advantage  of  this  representation  of  a 

reduced  residue  system  by  a  second  proof  of  the  generalized  form 

of  Wilson's  theorem  (Th.  21).     Let  p  be  an  odd  prime,  r  a  primi- 

1  Gauss :  Disq.  Arith.,  Arts.  57-93.  Dirichlet-Dedekind :  §§  127-131. 
Bachmann  :  Elemente  der  Zahlentheorie,  pp.  89-104.  Bachmann  :  Niedere 
Zahlentheorie,  pp.  322-348.    Mathews  :  §§  19-29.    Wertheim :  §§  48-69. 


THE   RATIONAL   REALM — CONGRUENCES.  105 

tive  root  of  p,  and  q1,q2,--,q4,<P)  any  reduced  residue  system, 
mod  p.  Since  the  integers  r,r2, -~,r<i>ip}  constitute  a  reduced 
residue  system,  mod  p,  each  of  the  q's  must  be  congruent  to  some 
one  of  these  powers  of  r,  mod  p ;  that  is, 


mod  p, 


where  llt  l2,  — ,J*c«  are  the  numbers  1,2,  —,♦(#)  in  some  order. 
Multiplying  these  congruences  together,  we  have 

But  r1+0(*»=r,  mod/', 

and  hence        tfi#2  **•>  2<mp>  — r  2    »  mod  p.  1) 

We  have  also 

r0<p>  —  1  =  (r^(^/2  —  1)  (V0W/2  +  1)  =0,  mod  p, 

and  hence,  since 

r0(p)/2 — 1^0,  mod/>, 

r  being  a  primitive  root  of  p, 

r0(p)/2  _|_  T  =  0j    m0(J  £#  2) 

Therefore  from  1)  and  2)  it  follows  that 

ftfa  •"$♦<#>  + 1  ssO*  mod/>. 
When  p  =  2,  this  proof  does  not  hold  as  </>(/>)  is  then  odd. 

§  30.    Indices. 

If  q  =  r*,  mod  />,  r  being  a  primitive  root  of  />  and  i  one  of  the 
numbers  o,  1,  •••,<£(/>)  —  x>  *  is  said  to  be  */*£  i«cte.r  0/  g  to  the 
base  r,  mod  p,  and  we  write  1=  indr  g,  mod  />. 

The  subscript  r  is  often  omitted,  in  which  case  it  is  understood  that 
all  indices  are  to  be  taken  to  a  certain  given  base. 

The  relation  of  an  integer  to  its  index  is  evidently  very  similar 


106  THE   RATIONAL   REALM CONGRUENCES. 

to  that  of  a  number  to  its  logarithm  and  indices  play  a  part  in  the 
theory  of  numbers  similar  to  that  of  logarithms  in  arithmetic.  It 
can  be  easily  shown  that  they  obey  the  following  laws : 

Let  p  be  the  modulus,  and  r  a  primitive  root  of  p. 

i.  The  index  of  the  product  of  two  integers  is  congruent  to  the 
sum  of  the  indices  of  the  factors,  mod  <f>(p),  that  is, 

indr  ab  ss  indr  a  +  indr  b,  mod  <£(/>). 

This  result  can  evidently  be  extended  to  the  product  of  any 
number  of  integers ;  that  is, 

indr  (cMV"  an)  =  indr  ax  +  indr a2  +  ""+  indr  a„,  mod <j>(p). 

ii.  The  index  of  the  nth  power  of  an  integer  is  congruent  to  n 
times  the  index  of  the  integer,  mod  <j>(p),  n  being  a  positive  in- 
teger; that  is, 

indr  an  =  n  indr  a,  mod  <f>(p). 

To  prove  i,  from  which  ii  at  once  follows,  let 

indr  a  =  ilt     indr  b  =  i2,     indr  ab  —  i. 

Then       assr*1,  modp,     Z?  =  rf2,  mod />,     ab^r1,  mod  p, 

and  hence  r4  =  r il+*2,  mod  p. 

Therefore  by  Th.  24  i  =  ix  -f  i2,  mod  <f>(p)  ; 

that  is  indr  ab  =  indr  a  +  indr  b,  mod  <f>  (p  ) . 

We  observe  that  in  every  system  indr  1=0.  By  means  of  the 
following  tables,  we  can  verify  these  results  and  illustrate  the  use 
of  indices.  Table  A  gives  for  the  modulus  13  the  index  to  the 
base  2  of  each  integer  of  a  reduced  residue  system,  and  Table  B 
gives  the  residue  corresponding  to  any  index  for  the  same  base 
and  modulus.  It  is  evident  that  two  integers  congruent  to  each 
other,  mod  p,  have  the  same  index  in  any  system  of  indices,  mod  p. 

Jacobi  has  given  in  his  Canon  Arithmeticus,  Berlin,  1839,  such  tables  for 
all  primes  less  than  1000.  See  also  for  such  tables  for  all  numbers  less 
than  100  that  have  primitive  roots  Wertheim,  Elemente  der  Zahlentheorie, 
also  Cahen  for  list  of  primitive  roots  and  tables  of  indices  for  every  prime 
number  less  than  200. 


THE   RATIONAL  REALM — CONGRUENCES. 
A. 


107 


Residue... 
Index 

1 

0 

2 

1 

3 
4 

4 
2 

5 
9 

6 

5 

7 
11 

8    1 
3 

9 

8 

10 
10 

11 

7 

12 

6 

B. 

Index 

Residue  ... 

0 
1 

1 
2 

2 
4 

3 
8 

4 
3 

5 
6 

6 
12 

t\ 

8 
9 

9 

5 

10 
10 

11 

7 

Ex.  Using  the  above  tables,  where  the  modulus  is  13  and  the  base  2, 
we  have  ind2  5  =  9,  ind2  9  =8. 

Therefore  ind245  ^ind25  +  ind29^  17,  mod  12,  and  hence  ind245  =  5. 
This  result  may  be  verified  by  observing  that 

45  =  6,  mod  13, 

whence  ind2  45  s  ind2  6,  mod  12 ; 

that  is,  ind245=  5. 

We  can  pass  from  a  system  of  indices  with  base  rlf  modp,  to 
one  with  the  base  r2  and  the  same  modulus  by  a  process  similar 
to  that  employed  in  passing  from  one  system  of  logarithms  to 
another. 

Let  p  be  the  modulus,  a  any  integer  not  divisible  by  p,  and 

f1  =  indr1  a,     i2  =  'mdra,    t='indr1  *V 

Then  we  have  a=^rj*,  mod/>, 

and  also  a  =  r2i2,  mod  p. 

But  r1^r2iJ  modp, 

and  hence  from  2)  and  3)  it  follows  that 

a  =  jy'S  modp, 

whence  indr  a  =  ii^  mod <f>(p)  ; 

that  is,  indr  a  =  indrg  ri '  ind^a,  mod  <f>(p). 

Therefore,  to  obtain  a  system  of  indices  to  the  base  r2  for  a  given 
modulus  p,  from  one  to  the  base  rx,  we  have  only  to  multiply  each 
index  of  the  latter  system  by  indT^rx  and  take  the  smallest  positive 
residue  of  the  products  with  respect  to  the  modulus  <f>(p). 
If  rlf  r2  be  any  two  primitive  roots  of  p,  then 

indrlr2-indr2rl^i,  mod<f>(p). 

This  follows  at  once  from  4)  by  putting  a  =  r2. 


2) 
3) 


4) 


io8 


THE   RATIONAL   REALM — CONGRUENCES. 


Ex.     To  obtain  for  the  modulus  13  a  system  of  indices  to  the  base  7 
from  one  to  the  base  2,  we  have  first  to  find  ind7  2. 

We  have  ind27-indT2^   1,  mod  12, 

and   from   table  A  ind27=  II, 

whence  nindT2^    1,  mod  12. 

Therefore  ind7  2  =  11. 

Multiplying  by  11  each  index  to  the  base  2  and  taking  the  least  positive 
residues  of  these  products  with  respect  to  the  modulus  12,  we  obtain  for 
the  modulus  13  the  following  system  of  indices  to  the  base  7. 


Residue... 
Index 


I 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

0 

II 

8 

10 

3 

7 

1 

9 

4 

2 

5 

12 

6 


Theorem  28.     //  indra,  mod  p,  be  i  and  d  be  the  greatest  com- 
mon divisor  of  i  and  p  —  I,  then  a  appertains  to  the  exponent 

{p-i)/d. 

We  have  a  =  r*,  mod  p. 

We  ask  what  is  the  smallest  value  of  m  for  which 

am  =  rmi=i,  mod  p.  5) 

By  Th.  24  we  must  have 

mi  =  o,  modp — 1, 


and  hence 


m 


p-l 


d  =  °>   m0d-d 


6) 


But  i/d  is  prime  to  (p  —  i)/d  and  (p — i)/d  is  therefore  the 
smallest  value  of  m  greater  than  zero,  that  will  satisfy  6).  Hence 
(p  —  i)/d  is  the  smallest  value  of  m  that  will  satisfy  5)  ;  that  is, 
a  appertains,  mod  p,  to  the  exponent  (p  —  i)/d. 

Cor.  If  r  be  a  primitive  root  of  p,  then  the  <f>(p  —  /)  primitive 
roots  of  p  are  those  (f>(p  —  1)  incongruent  powers  of  r  whose 
exponents  are  prime  to  p  —  1. 

Ex.  One  primitive  root  of  13  is  2.  Hence  the  4,  =  0(12),  primitive 
roots  of  13  are  2,  25,  27, 2U. 

§31.     Solution  of  Congruences  by  means  of  Indices. 
If  we  have  a  table  of  indices  to  any  base  for  a  given  modulus  p, 
we  can  solve  any  congruence  of  the  form 


THE   RATIONAL   REALM — CONGRUENCES.  IO9 

ax=b,  mod/>,  i) 

where  a  is  not  divisible  by  p ; 
for  from  i)  it  follows  that 

ind  a  -j-r  ind  x  =  ind  b,  mod  <f>(p), 
which  gives 

indjr==ind& —  ind  a,  mod  <f>(p), 

from  which  we  can  determine  ind  x  and  then  x. 

Ex.     From  the  congruence 

7x^4,  mod  13, 
» 
we  have  ind  x ^ ind  4  —  ind  7^2  —  11^  —  9,  mod  12. 

Hence  ind*  =  3, 

and  therefore  x^8,  mod  13. 

The  solution  of  the  congruence 

axn^b,  mod/>,  2) 

where  a  is  not  divisible  by  p,  can  be  reduced  by  the  use  of  indices 
to  the  solution  of  a  congruence  of  the  first  degree,  mod  <f>(p). 
For  from  2)  we  have 

ind  a  -f-  n'mdx='mdb,  mod  <f>(p), 
and  hence 

nind;r  =  indfr  —  ind  a,  mod  <f>(p),  3) 

that  is,  a  congruence  of  the  first  degree  in  the  unknown  ind  x. 
By  Th.  12  the  necessary  and  sufficient  condition  that  3)  shall  be 
solvable  is  that  indfr  —  ind  a  shall  be  divisible  by  the  greatest 
common  divisor,  d,  of  n  and  <f>(p).  When  this  condition  is  sat- 
isfied 3)  gives  \d\  values  of  ind;r,  corresponding  to  which  we  find 
\d\  values  of  x,  that  satisfy  2)  and  are  incongruent,  mod  p. 

In  the  following  examples  2  is  understood  throughout  to  be  the  base 
of  the  system  of  indices  employed,  tables  A  and  B  being  used. 
Ex.  1.     From  the  congruence 

$x~  ^4,  mod  13, 

we  have  7  ind x ^ ind 4  —  ind  5^2  —  9  ^  —  7,  mod  12. 

whence,  upon  removal  of  the  factor  '/,  that  is  prime  to  the  modulus  12, 

we  have  ind  .*•=== — 1,  mod  12. 


( 

IIO  THE   RATIONAL   REALM CONGRUENCES. 

Therefore  ind*  —  n, 

and  *s=7,  mod  13. 

Ex.  2.     From  the  congruence 

4*15  =  5,  mod  13,  4) 

we  have  15  ind  *  see  ind  5  —  ind  4  e=  9  —  2  ==7,  mod  12. 

The  greatest  common  divisor  of  15  and  12  does  not  divide  7.  Hence  4) 
has  no  roots. 

Ex.  3.     From  the  congruence 

x9  ==8,  mod  13, 

we  have  9  ind  x^  ind  8  ==3,  mod  12.  5) 

The  greatest  common  divisor  of  9  and  12  is  3  and  it  divides  the  second 
member,  3,  of  5).  Hence  5)  has  3  roots,  that  we  find  by  the  method 
of  Th.  12. 

From  5)  we  have  3  ind  x  ^  1,  mod  4, 

whence  ind*  ==3,  mod 4, 

and  consequently  ind  x^=  3,  7,  11,  mod  12. 

Therefore  ind*  =  3,7,  or  11; 

and  x^8, 11,  or  7,  mod  13. 

§  32.     Binomial  Congruences. 

The  subject  of  power  residues  and  in  particular  that  portion 
relating  to  primitive  roots  may  be  treated  from  another  point 
of  view,  that  of  the  binomial  congruence 

xn — is=o,  mod  p.1  1) 

We  see  by  §25  that  all  roots  of  1)  are  roots  of  the  congruence 

<\>{x)  =0,  mod/>, 

where  <j>{x)  is  the  greatest  common  divisor,  mo&p,  of  xn — 1 
and  x?-x — 1. 

It  is  easily  seen  that 

<f,(x)=xd—i, 

where  d  is  the  positive  greatest  common  divisor  of  n  and  p — 1. 

The  congruence 

xn — 1=0,  modp, 

^ahen:  p.  77.  Bachmann :  Niedere  Zahlentheorie,  p.  318.  H.  J.  S. 
Smith:    pp.  140-145. 


THE   RATIONAL   REALM CONGRUENCES.  Ill 

has  therefore  d  incongruent  roots,  that  are  the  roots  of 

xd — i==o,  mod/>.  2) 

We  can  now  confine  ourselves  to  congruences  of  the  form  2), 
where  d  is  a  divisor  of  p —  1. 

The  roots  of  1)  fall  into  two  classes,  those  which  satisfy  no 
congruence  of  the  same  form  and  of  lower  degree,  these  being 
called  primitive  roots,  and  those  which  satisfy  congruences  of 
this  form  and  of  lower  degree,  these  being  called  imprimi- 
tive  rootg. 

It  is  easily  seen  that  every  integer  that  is  a  root  of  a  con- 
gruence 

#* — 1^0,  mod/v,  3) 

where  d±  is  a  divisor  of  d,  is  also  a  root  of  2),  and  conversely  that 
every  imprimitive  root  of  2)  is  the  root  of  a  congruence  of  the 
form  3),  where  d±  is  a  divisor  of  d  smaller  than  d. 

The  primitive  roots  of  2)  are  evidently,  in  the  language  of 
power  residues,  those  integers  that  appertain  to  the  exponent  d, 
mod^.  They  are  evidently  <j>(d)  in  number  (Th.  2j).  The 
primitive  roots  of  p  are  the  primitive  roots  of  the  congruence 

xp-1  —  1=0,  mod  p. 

The  product  of  any  number  of  roots  of  2)  is  a  root  of  2)  and, 
in  particular,  any  positive  integral  power  of  a  root  of  2)  is  a 
root  of  2). 

If  r  be  any  primitive  root  of  2),  then  the  d  roots  of  2)  are 
by  Th.  23 

I,  r,  r2,  -",  r*-1. 

If  alf  a2  be  roots  of  the  congruences 

xdl  —  1=0,  mod  p,  3) 

and                                      x*2 — is==o,  mo&p,  4) 

respectively,  then  axa2  is  a  root  of  the  congruence 

j^id,  —  1=0,  mod  p.  5  ) 

In  particular,  if  alf  a2  be  primitive  roots  of  3)  and  4)  respect- 


112  THE   RATIONAL   REALM CONGRUENCES. 

ively  and  d1}d2  be  prime  to  each  other,  then  a±a2  is  a  primitive 
root  of  5)  (Th.  26). 

The  close  analogy  between  the  theory  of  binomial  congruences 
and  that  of  binomial  equations  will  be  easily  seen. 

§33.  Determination  of  a  Primitive  Root  of  a  Given  Prime 
Number.1 

The  method,  which  is  due  to  Gauss,  depends  upon  the  deter- 
mination of  a  series  of  integers  each  of  which  appertains  to  a 
higher  exponent  with  respect  to  the  given  prime,  p,  than  any  of 
the  preceding  ones. 

In  such  a  series  we  must  evidently  reach  an  integer  which 
appertains  to  the  exponent  p  —  1,  modp;  that  is,  which  is  a 
primitive  root  of  p. 

Take  any  positive  integer,  alf  less  than  p  and  greater  than  1, 
and  form  a  complete  system  of  its  power  residues,  modp. 

Let  us  suppose  that  ax  appertains  to  the  exponent  tlt  modp. 
If  tx  =  p — 1,  then  a±  is  the  primitive  root  required. 

If  t1=^=p  —  1,  it  is  evident  that  none  of  the  power  residues  of 
ax  Can  be  a  primitive  root  of  p,  for  they  are  the  roots  of  the 
congruence 

jfr  —  1  =  o,  mod  p,  1 ) 

and  hence  appertain,  modp,  to  exponents  not  greater  than  tv 

Suppose  that  t1=^=p  —  1.  We  proceed  to  determine  an  integer 
appertaining,  modp,  to  an  exponent  greater  than  tx.  Select  any 
positive  integer,  a2,  less  than  p  and  not  contained  among  the 
power  residues  of  alt  modp,  and  form  a  complete  system  of  its 
power  residues,  mod  p.  Let  t2  be  the  exponent  to  which  a2  apper- 
tains, modp.  If  t2—p  —  1,  a2  is  a  primitive  root  of  p  and  the 
problem  is  solved.  Suppose  that  t2=^=p  —  1;  then  t2  can  not  be 
a  divisor  of  tti  for  a2  would  in  that  case  be  a  root  of  the  con- 
gruence 1)  and  hence  a  power  residue  of  alf  modp,  which  is 
contrary  to  our  hypothesis. 

If  t2  be  a  multiple  of  ^  but  =%=p —  1,  we  have  found  an  integer, 

1  Gauss  :  Disq.  Arith.,  Art.  73  Ca'.ien :  pp.  90-95.  Mathews  :  pp.  20-22. 
H.  J.  S.  Smith :   pp.  49-54. 


THE   RATIONAL   REALM CONGRUENCES.  II3 

a2,  appertaining  to  a  higher  exponent  than  alt  modp,  although 
not  a  primitive  root  of  p.  We  then  select  a  positive  integer  less 
than  p  and  not  contained  among  the  power  residues  of  a2,  form 
its  power  residues,  modp,  and  proceed  as  before.  Suppose,  how- 
ever, that  t2  is  not  a  multiple  of  tls  and  let  m  be  the  least  common 
multiple  of  tx  and  t2.  It  is  evident  that  m  is  greater  than  tlf 
since  t2  is,  not  a  divisor  of  tv  We  shall  show  how  to  determine 
an  integer  appertaining  to  the  exponent  m,  mod  p. 

We  first  resolve  m  into  two  factors,  mlftn2,  prime  to  each 
other  and  divisors  of  tx  and  t2  respectively.  This  may  be  accom- 
plished as  follows. 

Let  px  be  a  prime  that  occurs  to  the  power  e1  as  a  factor  of  tx 
and  to  the  power  e2  as  a  factor  of  t2.  We  take  pxei  as  a  factor  of 
mlf  or  pxe2  as  a  factor  of  w2,  according  as  ex  is  greater  or  less 
than  e2.  If  e1  =  e2,  then  /^i  may  be  taken  as  a  factor  of  either 
mt  or  w2.  We  have  then  m  =  m1m2  =  t1/d1-t2/d2,  where  dls  d2 
are  respectively  the  product  of  primes  that  occur  in  the  case  of 
dx  to  a  lower  power  in  tx  than  in  t2,  and  in  the  case  of  d2  to  a 
lower  power  in  f2  than  in  t±. 

Consider  now  the  residues,  modp,  of  a/1,  and  a/2.  These 
integers  appertain  respectively  to  the  exponents  tjdx,  tjd2,  that 
are  prime  to  each  other. 

Hence  their  product  axdia2d2  appertains  to  the  exponent  m,  that 
is  the  product  of  these  exponents  (Th.  26). 

Ex.  To  find  a  primitive  root  of  157.  The  power  residues  of  2,  mod 
157,  are 


2, 

4, 

8, 

16, 

32, 

64,       128, 

99, 

4i, 

82, 

7, 

14, 

28, 

56, 

112, 

67,       134, 

in, 

65, 

130, 

103, 

49, 

98, 

39, 

78, 

156, 

—  2, 

—  4, 

—  8, 

-16, 

—  32, 

—  64,  —128, 

—  99, 

—  4i, 

-82, 

—  7, 

—  14, 

—  28, 

-56,- 

—  112, 

—  67,  —134,- 

—  in, 

-65, 

—  130, 

-103, 

—  49, 

-98, 

—  39, 

-78, 

-iS6asi. 

The  work  is  shortened  by  observing  that  the  residue  of  228  is  — 1,  and 
consequently  the  remaining  26  residues  are  the  negatives  of  the  first  26. 
We  see  that  2  appertains  to  the  exponent  52,  mod  157.  The  integer  3, 
not  being  contained  among  the  residues  of  2,  we  form  its  power  residues, 
mod  157,  and  find  that  it  appertains  to  the  exponent  78. 
8 


114  THE   RATIONAL   REALM CONGRUENCES. 

We  have  52  =  22-i3, 

and  78  =  2-3-13. 

The  least  common  multiple  of  52  and  78  is  156,  that  can  be  resolved  into 
two  factors  prime  to  each  other  and  divisors  of  52  and  78  respectively. 

Thus  ^^x'-^-^^X^. 

0  13  2  13       2 

The  integers  213  and  32  appertain  to  the  exponents  —  and  —  respectively, 

13  2 

and   hence   their  product  21332  appertains   to  the   exponent   156;   that   is, 
21332  is  a  primitive  root  of  157.     But  we  have  seen  that 

213==28,  mod  157. 

Hence  213-32==28-9  =  252==55,  mod  157. 

We  have  therefore  55  as  a  primitive  root  of  157. 

We  could  have  resolved  156  in  another  way,  since  13  occurs  to  the  same 
power  in  52  and  78. 

Thus  I56  =  ^I3xi^_L3  =  5£xZ? 

J  I         ^      2  •  13  I    /X  26 

Then  2  and  326  appertain  to  the  exponents  52  and  3  respectively,  and 
their  product  2-3M  appertains  to  the  exponent  156;  that  is,  2-328  is  a 
primitive  root  of  157. 

We  have  2-3*^ 2- 144 ^288^  131,  mod  157, 

and  hence  131  is  a  primitive  root  of  157.    For  this  example  and  a  table  of 
the  power  residues  of  55,  mod  157,  see  Cahen :    pp.  92,  93. 

§34.    The  Congruence  xHE=b,  mod  p.     Euler's  Criterion. 
The  congruence 

C^j^essfr^  modp, 

where  ax  is  not  divisible  by  p,  can  always  be  reduced  to  the  form 

xn  =  b,  modp, 

and  in  this  form  it  has  a  special  interest.     In  what  follows  we 
consider 

b^o,  mod  p. 

From  what  has  been  said  in  §31,  the  truth  of  the  following 
theorem  is  at  once  evident. 

Theorem  29.     The,  necessary  and  sufficient  condition  that  the 

congruence  xn^=b,  modp,  1) 


THE   RATIONAL   REALM CONGRUENCES.  I  I  5 

shall  be  solvable,  is  that  ind  b  shall  be  divisible  by  the  greatest 
common  divisor,  d,  of  n  and  <f>(p)  ;  this  condition  being  satisfied 
the  congruence  has  exactly  \d\  incongruent  roots. 

See  §  31,  Ex.  3. 

Since  ind"&  varies  with  the  primitive  root  taken  as  base  of 
the  system  of  indices  used,  this  condition  for  the  solvability  of 
1)  appears  to  depend  upon  the  primitive  root  selected. 

It  is  evident,  however,  that  in  reality  the  solvability  of  i)  is 
in  no  way  dependent  upon  this  selection,  and  it  must  be  possible 
therefore  to  find  a  criterion  for  the  solvability  of  this  congruence 
that  is  independent  of  indices. 

Such  a  criterion  is  that  first  given  by  Euler  and  known  as 
Euler's  criterion.     It  is  contained  in  the  following  theorem. 

Theorem  30.  //  d  be  the  positive  greatest  common  divisor 
of  n  and  <f>(p),  the  necessary  and  sufficient  condition  that  the 

congruence  xn  =  b,  mod  p,  2) 

shall  be  solvable  is       b4>w/d=i,  mod  p.  j) 

This  condition  being  satisfied,  the  congruence  has  exactly  d  incon- 
gruent roots. 

Let  r  be  any  primitive  root  of  p,  and  let 

indr  b  =  c. 

Suppose  2)  to  be  solvable,  then  c  is  divisible  by  d. 

Let  c  =  md. 

Then  b^rmd,  mo&p, 

and  btw/d^rm-tw,  modp, 

whence  b<pw/d=  I}  mod/>. 

Therefore  3)  is  a  necessary  condition  for  the  solvability  of  2). 
Conversely,  if  b  satisfy  3),  the  index  of  b  in  every  system  of 
indices,  mod  p,  must  be  divisible  by  d ;  for,  if 

b==rc,  modp, 
then  b<t>(pWd  =  rc<f>(P)/d}  mod/>, 


Il6  THE   RATIONAL   REALM CONGRUENCES. 

and  hence  rc0d»/tf==  I?  modp. 

Since  r,  being  a  primitive  root  of  p,  appertains  to  the  exponent 
<f>(p),  c<f>(p)/d  must  be  divisible  by  <\>{p). 

Therefore  c/d  is  an  integer ;  that  is,  c  is  divisible  by  d.  Hence 
3)  is  a  sufficient  as  well  as  necessary  condition  for  the  solvability 
of  2).  That  the  congruence  when  solvable  has  d  roots  is  evident 
from  the  preceding  paragraph. 

All  incongruent  integers  b,  for  which  the  congruence  2)  is 
solvable  may  be  obtained  by  observing  that  they  are  the  roots  of 
the  congruence 

,r*<9>/*sBi,  mod  p.  4) 

This  congruence  has  <f>(p)/d  incongruent  roots,  since  <f>(p)/d 
is  a  divisor  of  <f>(p).  These  roots  are  the  incongruent,  modp, 
values  of  b  for  which  2)  is  solvable.  Such  numbers  congruent 
to  the  nth  power  of  an  integer,  modp,  are  called  the  «-ic  resi- 
dues of  p,  and  we  have  the  following  theorem. 

Theorem  31.  The  number  of  incongruent  n-ic  residues, 
mod  p,  is  <f>(p)/d,  where  d  is  the  positive  greatest  common  divisor 
of  n  and  <f>(p),  and  these  residues  are  the  rodts  of  the  congruence 

x<t>w/d==i,  modp. 

Thus,  if  p  =  7,  we  have  for 

11  =  2,  3  incongruent      quadratic  residues  of  7, 

w  =  3, 2  incongruent  cubic  residues  of  7, 

n  =  4,  3  incongruent  biquadratic  residues  of  7, 

m  =  5,6  incongruent  quintic  residues  of  7, 

n  =  6, 1  incongruent  sextic.  residue    of  7, 

and  so  on. 

We  may  obtain  the  above  results  and  also  the  residues  them- 
selves by  raising  each  number  of  a  reduced  residue  system,  mod  p, 
to  the  nth  power  and  determining  the  number  of  the  reduced 
residue  system  to  which  each  of  these  nth  powers  is  congruent, 


THE   RATIONAL  REALM — CONGRUENCES.  II7 

mod  p.     Thus  for  p  =  7,  we  take  as  a  reduced  residue  system 
1,  2,  3,  4,  5,  6,  and  have  for 

n  =  2,  i2  =  i,  22  =  4,  32  =  2,  42  =  2,  52  =  4,  62^i/ 

w  =  3,  i3  =  i,  23e==i,  33  =  6,  43=i,  53  =  6,  63  =  6, 

n  =  4,  i4=i,  24  =  2,  34  =  4,  44  =  4,  54  =  2>  64=i,  [.mod 7. 

w  =  5,  i5=i,  25  =  4,  35  =  5.  45  =  2,  55  =  3,  65  =  6, 

n  =  6,  i'bbi,  26=i,  36=i,  46=i>  56  =  i>  66=i, 

Hence  the  incongruent  quadratic  residues  of  7  are  1,  2  and  4, 
the  cubic  residues  1  and  6,  the  biquadratic  residues  1,  2  and  4,  the 
quintic  residues  1,  2,  3,  4,  5  and  6,  the  sextic  residue  1. 

An  integer  is  therefore  a  quadratic  residue  of  7  when  and  only 
when  it  is  congruent  to  one  of  the  integers  1,  2,  4,  mod  7,  and 
likewise  for  the  other  values  of  n. 

In  the  next  chapter  we  shall  discuss  fully  the  subject  of  quad- 
ratic residues. 

Investigations  concerning  the  properties  of  cubic  and  biquad- 
ratic residues  have  led  to  important  developments  in  the  theory 
of  numbers,  that  will  be  noticed  later. 

Examples. 

1.  Show  that  ,r13  —  x  is  divisible  by  2730,  x  being  any  integer. 

2.  If  x  be  a  prime  greater  than  13,  x12 —  1  is  divisible  by  21840. 

p(p-i) 

3.  If  p  be  a  prime  and  a  prime  to  p,  then  either  a    2    —  1  or 

a    2     +  1  is  divisible  by  p2. 

4.  No  number  of  the  form  m4  -f-  4  except  5  is  prime. 

5.  The  product  of  numbers  of  the  form  mx  +  1  is  a  number 
of  the  same  form. 

6.  The  cube  of  any  integer  not  divisible  by  3  is  congruent  to 
±  1,  mod  9. 

7.  Solve  the  congruences 

a)  x3 —   &r+    1=0,  mod  5. 

b)  x4  +  6x3  —   8x2  +  lyc  +    5  =  o,  mod  7. 

c)  x*-\-2x3 — i3.F2-f    Sx  +  x3  =  0>  mod  11. 

8.  The  congruence 


mod  15. 


I  1 8  THE   RATIONAL   REALM CONGRUENCES. 

8.r5  +  4-r4  —  3,i-3  -f-  3X2  +  3^  +  6  =  o,  mod  7, 

has  a  multiple  root;   solve  the  congruence. 

9.  Solve  the  system  of  congruences 

3*  —  43'+    52  —  911  =    1 

2*  +  $y+   4^  +  5^=   8 

■*"  +  53'  +   6z-{-2u=   1 

7X  —  33'  —  I02  +  2U  ■■  I0 

10.  Solve  the  congruence 

a-5  —  &r4  +  5jt3  —  5-r2  +  4.1-  -j-  3  =  o,  mod  27. 

11.  Solve  the  congruence 

x5  —  6.r4  +  8a-3  —  4*2  +  jx  -f  2  =  0,  mod  20. 

12.  Prove  Th.  30  without  the  use  of  indices. 

13.  Find  the  prime  polynomials  of  the  third  degree,  mod  5. 

14.  If  a  appertain  to  the  exponent  ta,  mo&p,  then 

1  -f  a  +  a2  -f  •  •  •  +  a**-1  s==  o,  mod  p, 

(Gauss:  Disq.  Arith.,  Art.  79.) 

15.  The  product  of  all  incongruent  primitive  roots,  mo&p,  is 
congruent  to  1,  mod/?,  except  when  p  =  3.  {Ibid.:  Art.  80.) 

16.  If  rlt  r2,  •••,r^m)  be  a  reduced  residue  system,  modw,  then 
all  primes  are  contained  in  the  forms 

km  +  rlf  km  +  r2,  •  •  • ,  km  +  rUm). 

17.  If  p  be  a  prime  of  the  form  \n —  1  and  a  appertain,  mod/>, 
to  the  exponent  (p  — 1)/2,  then  — a  is  primitive  root  of  p. 

18.  Use  theorem  in  Ex.  17  to  determine  a  primitive  root  of  191. 
(Cahen:   p.  94.) 

19.  Determine  a  primitive  root  of  73  (Gauss:  Disq.  Arith., 
Art.  74),  also  one  of  97  (Mathews :  p.  20). 

20.  If  p  be  a  prime  and  rx,r2,  •••,^(P)  a  reduced  residue  sys- 
tem, mod/>,  every  rational  integral  symmetric  function  of  the 
r's,  whose  degree  is  not  a  multiple  of  (f>(p),  is  divisible  by  p. 
(Cahen:   p.  109.) 

21.  Solve  the  congruences 

a)  x20  =   3,  mod  13. 

b)  ,r9  =  io,  mod  13. 


CHAPTER   IV. 

The  Rational  Realm, 
quadratic  residues.1 
§  i.    The  General  Congruence  of  the  Second  Degree  with  One 
Unknown. 

The  most  general  congruence  of  the  second  degree  with  one 
unknown  has  the  form 

ax2  +  bx  +  c  —  °>  m°d  m.  I ) 

We  have  seen  (Chap.  Ill,  $2j)  that  the  solution  of  i)  when  m 
is  a  composite  number  can  be  reduced  to  the  solution  of  a  system 
of  congruences  of  the  same  form  but  with  prime  moduli.  We 
shall  therefore  confine  ourselves  to  the  case  in  which  m  is  a  prime 
number,  p,  and  furthermore,  since  for  p  =  2  the  congruence  is 
easily  solvable  by  trial,  we  shall  suppose  p  odd. 
We  consider  then  the  congruence 

*  ax2  +  bx  -\-  c  =  o,  mod  p,  2) 

where  a  is  not  divisible  by  the  odd  prime  p,  for  if  it  were,  the  con- 
gruence would  not  be  of  the  second  degree.  Multiplying  2)  by 
the  reciprocal,  at,  mod  p,  of  a,  we  obtain  the  congruence 

x2  +  axbx  -\-  axc  ==  o,  mod  p.  3) 

If  now  the  coefficient  of  x  in  3)  be  not  even,  we  make  it  so  by 
putting  aj?  +  P  for  (hf>.  Having  done  this,  if  necessary,  3)  is 
transformed  into  the  equivalent  congruence 

x2  +  2btx  -f-  cx  =5  o,  mod  p.  4) 

Adding  b±2  to  both  members  of  4),  we  obtain 
(x-{-b1)2  =  b12  —  clt  modp, 
or  putting  x  +  bx  =  s,  mod  />,  5) 

&x2  —  cx^d,  mod/', 

1  Gauss:  Disq.  Arith.,  pp.  73-119.  Wertheim:  pp.  170-236.  Cahen:  pp. 
113-143.  Bachmann:  Niedere  Zahlentheorie :  pp.  180-317.  Dirichlet- 
Dedekind:    pp.  75-127. 

119 


120  QUADRATIC   RESIDUES. 

we  see  that  the  solution  of  2)  can  be  reduced  to  the  solution  of  a 
binominal  congruence 

z-^d,  mod/>.  6) 

If  d^o,  mod/0  7) 

the  congruence  6)  has  either  no  roots  or  two  incongrnent  roots, 
for  if  r  be  a  root,  then  —  r  is  also  a  root,  and  if 

r==  —  r,  modp, 

then  2r==o,  mo&p, 

and  hence  r  =  o,  mod/>, 

which  is  impossible  from  7). 

The  solutions  of  4),  or  what  is  the  same  thing  3),  being  con- 
nected with  those  of  6)  by  the  relation  5),  we  see  that  4)  has  two 
incongruent  roots  or  no  roots  according  as  6)   has  two  incon- 
gruent  or  no  roots. 
If  d  =  o,  mo&p, 

then  6)  has  the  two  equal  roots 

£  =  0,  modp, 

and  4)  has  the  two  equal  roots1 

#h==  —  b19  modp,  1 

x2  +  2b±x  +  C\  being  a  perfect  square,  mod  p.  The  solutions  in 
the  case  of  equal  roots  being  obvious,  we  shall  exclude  this  case 
and  confine  ourselves  therefore  to  the  consideration  of  binomial 
congruences  of  the  form  6),  where 

c?HJ=o,  mod  p. 

The  analogy  shown  here  between  quadratic  equations  and  congruences 
of  the  same  degree  with  prime  modulus  should  be  noticed,  the  vanishing 
of  the  discriminant  b2  —  \ac  of  ax2-\-bx-{-  c  being  in  the  one  case  the 
condition  that  the   equation 

ax1  +  bx  4-  c  =  o, 

shall  have  equal  roots,  and  the  divisibility  of  b2  —  4ac  by  the  modulus 
being  in  the  other  case  the  condition  that  the  congruence 

1  Wertheim:  p.  170. 


♦  QUADRATIC   RESIDUES.  121 

ax2  +  bx  -f-  c  ra  o,  mod  />, 

shall  have  equal  roots. 

Ex.    Let  5*2 — \ix  —  12  =  0,  mod  23, 

be  the  proposed  congruence.  Multiplying  it  by  14,  the  reciprocal,  mod 
23,  of  5,  we  obtain  the  equivalent  congruence. 

70X2  —  154* — 168  ==0,  mod  23, 

or  x2  —  16*  —  7  ==  o,  mod  23, 

or  (x  —  8)2==2,  mod  23. 

Putting  x  —  8  ==2,  mod  23,  8) 

we  have  ^^2,  mod  23, 

which  has  the  roots  z  as  5  or  —  5,  mod  23. 

These  substituted  in  8)  give  the  two  roots  of  the  original  congruence 

x^i$  or  3,  mod  23. 

§  2.    Quadratic  Residues  and  Non-residues. 

An  integer,  a,  prime  to  the  modulus  m.  is  said  to  be  a  quadratic 
residue  or  non-residue  of  m,  according  as  the  congruence 

„v2==a,  mod  wi, 

has  or  has  not  roots;  that  is,  a  is  said  to  be  a  quadratic  residue  of 
m,  if  it  be  a  residue,  mod  m,  of  some  square  number,  and  a  quad- 
ratic non-residue  of  m,  if  it  be  a  residue,  mod  m,  of  no  square 
number. 

Ex.  1.    The  congruence        x^^2,  mod  7, 
has  the  roots  3  and  —  3 ;  hence  2  is  a  quadratic  residue  of  7. 

Ex.  2.    The  congruence       x2^s,  mod  7, 

has  no  roots,  as  may  be  seen  by  trying  the  integers  —  3,  —  2,  —  1, 
o,  1,  2,  3  (also  see  Chap.  Ill,  §  34)  ;  hence  5  is  a  quadratic  non-residue 
of  7. 

If  there  be  no  danger  of  misunderstanding,  the  word  quadratic 
is  omitted.  The  behavior  of  the  integer  a  in  this  relation  is  called 
its  quadratic  character  with  respect  to  the  modulus  m.  It  is  evi- 
dent that  all  integers  belonging  to  the  same  residue  class,  mod  m, 
have  the  same  quadratic  character  with  respect  to  m.  We  have 
now  two  principal  questions  to  answer  concerning  the  congruence 

x2?==a,  modw. 


122  QUADRATIC    RESIDUES. 

I.  What  integers  are  quadratic  residues  of  a  given  modulus  m? 

II.  Of  what  moduli  is  a  given  integer,  a,  a  quadratic  residue? 
We  shall  confine  ourselves  now  to  the  case  in  which,  m  is  a 

prime,  p.  Furthermore,  we  may  suppose  p  to  be  odd,  since  the 
case  p  =  2  is  at  once  disposed  of  by  observing  that  all  odd  integers 
are  quadratic  residues  of  2,  and  all  even  integers,  being  not  prime 
to  2,  are  excluded  from  consideration.  For  convenience,  we  also 
suppose  p  to  be  positive. 

We  have  as  a  special  case  of  Th.  30,  Chap.  Ill,  the  following : 

Eule/s  Criterion. 
Theorem  i.     The  necessary  and  sufficient  condition  that  a  shall 
be  a  quadratic  residue  of  p;  that  is,  that  the  congruence 

x2  =  a,  mod/>, 

shall  have  roots,  is        a(p_1)/2=i,  mod  p.    - 

Cor.  1.  The  integer  a  is  a  quadratic  residue  or  non-residue  of 
p  according  as  we  have 

a<p-i>/*  ==  1,  or  —  1,  mod  p  ; 

for  since  aP~x  —  1=0,  mod  p, 

then  (aW2  —  1 )  (a^/2  +  1 )  =  o,  mod  p  ; 

whence  it  follows  that  either 

a(p-D/2_I==0>  mod/), 

or  a(P-1)/2  _|_  1  sb  o,  mod  p. 

-Therefore  if  a(P-1)/2==  1,  mod  p,  a  is  a  quadratic  residue  of  p,  and 
if  a(P-1)/2== —  i}  mod  p,  a  is  a  quadratic  non-residue  of  p. 

Cor.  2.  The  product  of  two  quadratic  residues  or  of  two  quad- 
ratic non-residues  of  p  is  a  quadratic  residue  of  p,  and  the  product 
of  a  quadratic  residue  and  a  quadratic  non-residue  of  p  is  a  quad- 
ratic non-residue  of  p. 

Let  alf  a2  be  quadratic  residues,  and  a3,  a4  quadratic  non-residues 
of  p. 

Then  since  a1(p_1)/2^  1,  modp, 


QUADRATIC    RESIDUES.  I  23 

and  a*<**>/**arl,  mod  p, 

it  follows  that       .      O^) (p~1)/2  =  I,  mod  p. 

Hence  axa2  is  a  quadratic  residue  of  />. 

Since  '    fl3<p-i)/2=_  1,  mod/', 

and  a/p-1)/2^— 1,  mod/>, 

it  follows  that  Os^) (p~1)/2  ■■  I,  mod  p. 

Hence  a3a4  is  a  quadratic  residue  of  />. 

Since  ^^/'a  I,  mod/>, 

and  ^OHiVS  =  —  i,  mod  />, 

it  follows  that  O^g)  **>/*■»—  i,  mod  p. 

Hence  axa3  is  a  quadratic  non-residue  of  />.  From  Cor.  2  follows 
at  once : 

Cor.  3.  The  product  of  several  integers  is  a  quadratic  residue 
or  non-residue  of  p,  according  as  an  even  or  odd  number  of  the 
integers  are  quadratic  non-residues  of  p. 

It  is  therefore  only  necessary  to  be  able  to  determine  the  quad- 
ratic character  of  all  prime  numbers  with  respect  to  any  modulus  p. 

Ex.  1.  **eB3,  mod  13.  1) 

We  have  3(13_1)  /2  =  36  =  I,  mod  13. 

Hence  3  is  a  quadratic  residue  of  13,  the  roots  of  1)  being  4  and  — 4. 
Ex.  2.  ;Te=7,  mod  13. 


Hence  7  is  a  quadratic  non-residue  of  13. 

We  can  verify  the  result  by  substituting  the  numbers,  ±1,  ±2,  ±3, 
±  4,  ■  ±  5,  ±6,  which  give 


I £7      9¥k7    25^7    I     modl3. 
4=^7     16^7    36=£7  j 


This  also  follows  from  the  fact  that  ind2  7,  mod  13,  is  not  divisible  by  2. 

Ex.  3.     Since  21  =  3-7 

and  3  is  a  residue  of  13,  and  7  a  non-residue  of  13,  21  is  by  Cor.  2  a  non- 
residue  of  13,  which  is  verified  by 

2I(13-1)/2_(_5)6)     modl3> 

^((-5)2)3^(-i)3^-i.  mod  13. 


124  QUADRATIC   RESIDUES. 

§  3.?  Determination  of  the  Quadratic  Residues  and  Non- 
residues  of  a  Given  Odd  Prime  Modulus. 

Theorem  2.  //  p  be  an  odd  prime,  one  half  the  integers  of  a 
reduced  residue  system,  mod  p,  are  quadratic  residues  of  p}  and 
the  other  half  non-residues. 

First  Proof: 

Take  as  a  reduced  residue  system,  mod  p,  the  integers 

p-i        p-3  _2_I2.       P-I  P-*      _x 

2  2  2  2  ' 

The  squares  of  the  integers 

2  2 

are  incongruent  each  to  each,  mod  p,  for  if  (p  —  r)/2  and 
(p  —  s)/2  be  any  two  of  these  integers,  r  and  s  being  integers 
of  the  series  1,3,  •••,/>  —  2,  and  unequal,  and 

m.     (^r+ti)(^-^)-o.„odA 

whence  either 1- =  o,  mod  p,  3) 

2  2 

or  P^ll  —  t—L  m  o,  mod  py  4) 

2  2 

Both  3)  and  4)  are,  however,  impossible,  since  (p  —  r)/2  and 
(p  —  s)/2  are  unequal  and  both  positive  and  less  than  p/2. 

The  squares  of  the  -J  (p  —  1)  integers  2)  give,  therefore,  -J (p  —  1) 
incongruent  residues,  mod  p,  and  these  are  all  the  incongruent 
quadratic  residues  of  p}  for  the  squares  of  the  remaining  integers 
of  1)  give  evidently  the  same  residues. 

Hence  the  theorem. 


QUADRATIC   RESIDUES.  125 

Second  Proof: 

Let  r  be  a  primitive  root  of  p.     Then 

r,r2,  '■-,ri,  •••,rP-1 

is  a  reduced  residue  system,  mod  p. 

From  Chap.  Ill,  Th.  29,  it  follows  at  once  that  every  power  of 
r  with  an  even  exponent  is  a  residue  of  p,  and  every  power  of  r 
with  an  odd  exponent  is  a  non-residue. 

Hence  there  are  J(/> — 1)  residues  of  p  and  \{p — 1)  non- 
residues  of  p. 

We  can  express  this  also  by  saying  that  those  of  the  integers  of 
a  reduced  residue  system  which  have  even  indices  are  residues  of 
p,  while  those  which  have  odd  indices  are  non-residues.  The 
residues  of  any  prime  for  which  we  have  a  table  of  indices  can 
evidently  be  easily  thus  determined. 

Th.  1,  Cor.  2,  can  be  deduced  from  the  second  proof  given 
above  in  a  very  elegant  manner ;  for  if 

a^=axa2  •••  an, 

then  ind  a  =  ind  ax  +  ind  a2  -f-  •  •  •  -f-  ind  an,  mod  <j>(p), 
and  hence,  since  <f>(p)  is  even,  ind  a  is  odd  or  even  according  as 
ind  ax  -f-  ind  a2  -f-  •  •  •  -j-  ind  an  is  odd  or  even.  But  ind  ax  -j-  ind  a2 
-f-  •  •  •  -f-  ind  an,  and  hence  ind  a,  is  odd  or  even  according  as  an  odd 
or  even  number  of  the  indices  of  ax,a2,  ••-J(in  are  odd.  Hence  a 
is  a  quadratic  residue  or  non-residue  of  p  according  as  an  even  or 
odd  number  of  its  factors  a^,a2,  "-,an  are  quadratic  non-residues 
of  p. 

We  can  now  answer  fully  the  first  of  our  two  questions  con- 
cerning the  congruence 

#2  =  a,  modpj 

where  p  is  an  odd  prime ;  for  suppose  that  we  have  any  reduced 
residue  system,  mod  p,  and  that  those  residues  of  this  system 
which  are  quadratic  residues  of  p,  are  rlr  r2,  ••-,r^(i,)  and  those 
which  are  quadratic  non-residues  of  p  are  n1}  n2,  '",n^(P),  this 
having  been  determined  by  any  of  the  methods  given  above.  Then 
all  those  and  only  those  integers  included  in  the  forms 
kp  +  rlf  kp  +  r2,--.,  kp  +  rmp) 


126  QUADRATIC   RESIDUES. 

are  quadratic  residues  of  p,  and  all  and  only  those  integers  included 

in  the  forms       kp  +  nlt  kp  -f-  n2,  •••,  kp  +  w^(P) 

\  are  quadratic  non-residues  of  p,  k  taking  all  integral  values. 
\     Ex.  i.    Let  p  =  17,  and  take  as  a  reduced  residue  system, 

—  8,  —7,  —6,  —5,  —4,  —3,  —2,  —1,  1,  2,  3,  4,  5,  6,  7,  8 
We  have 

(±i)2=l,  (±3)a=  9,  (±5)'— 81  (±7)^15,-1   ^od 


(±2)2  =  4,  (±4)!=i6,  (±6)2  =  2,  (±8): 


.5,-1 
13./ 


Hence  1,  2,  4,  8,  9,  13,  15,  16  are  the  incongruent  quadratic  residues 
of  17,  and  all  those  and  only  those  integers  which  are  included  in  the  forms 
17k  +  1,  17k  +  2,  17&  +  4,  I7&  +  8,  I7&  +  9,  17^  +  13,  I7H-  15,  17^  +  16, 
are  quadratic  residues  of  17. 

The  incongruent  quadratic  non-residues  of   17  are 

3,  5,  6,  7,   10,   11,  12,  14, 

and  hence  all  and  only  those  integers  which  are  included  in  the  forms 
*7*  +  3,  Wk  +  5,  vjk  +  6,  176  +  7,  17*  +  10,  17k  -f  11,  17&  +  12,  17&  +  14, 
are  quadratic  non-residues  of  17. 
Ex.  2.    Let  p  as  13. 

From  table  A,  Chap.  IV,  §  30,  we  see  that  the  indices  of  1,  3,  4,  9, 
10  and  12  are  even,  and  the  indices  of  2,  5,  6,  7,  8  and  11  are  odd. 

Hence  1,  3,  4,  9,  10  and  12  are  the  incongruent  quadratic  residues  of 
13,  and  2,  5,  6,  7,  8,  and  11  are  the  incongruent  quadratic  non-residues 
of   13. 

We  see  then,  as  above,  that  the  quadratic  residues  of  13  are  integers 
of  the  forms 

I3&  +  Xi  I3&  +  3,  Uk  +  4,  13^  +  9,  ]3k  +  10,  13^  -f- 12, 
and  the  quadratic  non-residues  of   13  of  the  forms 

13&  +  2,  I3&  +  5,  13^  +  6,  i3&  +  7>  13^+8,  13&  +  H. 
We  have  now  answered  fully  the  first  question  concerning  the 
congruence  x2  =  a,  mod  p  ; 

that  is,  we  are  able,  as  shown  in  the  two  examples  above,  to  give 
for  any  value  of  p  a  finite  system  of  forms,  kp  -f-  r,  where  r  is  a 
known  integer  and  k  any  integer,  such  that  all  and  only  those 
integers  obtained  from  these  forms  by  letting  k  take  all  integral 
values,  are  quadratic  residues  of  p. 

A  similar  series  of  forms  may,  as  was  shown  above,  be  given 
for  the  non-residues  of  p. 


QUADRATIC   RESIDUES.  \2J 

Before  considering  the  second  question,  that  is,  of  what  odd 
prime  moduli  is  a  a  quadratic  residue,  we  shall  introduce  a  sym- 
bolic notation  which  will  greatly  simplify  the  discussion. 

§  4.    Legendre's  Symbol. 

The  quadratic  character  of  an  integer  a  with  respect  to  a  prime 
p,  can  be  expressed  in  a  very  convenient  manner  by  means  of  the 
following  symbol  introduced  by  Legendre. 

Let  (a/p)  denote  -f-  1  or  —  1,  according  as  a  is  a  quadratic 
residue  or  non-residue  of  p ;  that  is,  (a/p)  =  i  denotes  that  a 
is  a  quadratic  residue  of  p  and  (a/p)  = — 1  denotes  that  a  is  a 
quadratic  non-residue  of  p.  In  what  follows,  p  is  assumed  first 
of  all  to  be  odd,  and  secondly,  for  the  sake__of  convenience,  posi^ 
tive.  This  last  assumption  is  not  necessary,  but  simply  to  avoid 
the  trouble  of  writing  \p\  when  the  absolute  value  of  p  is  to  be 
taken.     Combining  this  with  Euler's  criterion,  we  see  that 


G)-«- 


mod/, 


expresses  the  quadratic  character  of  a,  with  respect  to  p. 
From  Th.  1,  Cor.  3,  it  is  evident  that 


(^)-(l)G)-G) 


If  a=  by  mod  /, 

then 


Also 


G)-G> 


denotes  that  the  quadratic  character  of  a  with  respect  to  p  is  the 
same  as  the  quadratic  character  of  b  with  respect  to  p,  and 

G)~GM*)G)~  ■ 


128  QUADRATIC   RESIDUES. 

denotes  that  the  quadratic  character  of  a  with  respect  to  p  is 
opposite  to  the  quadratic  character  of  b  with  respect  to  p. 
If  a  =  k2aly  then  since  (k2/p)  =  i, 


(>)-(')  (5) -(5> 


In  determining  the  value  of  (a/p)  we  may  therefore  suppose 
all  square  factors  to  have  been  removed  from  a. 

§  5.  Determination  of  the  Odd  Prime  Moduli  of  which  a 
Given  Integer  is  a  Quadratic  Residue. 

To  answer  the  second  question :  of  what  odd  prime  moduli  is  a 
a  quadratic  residue,  of  what  a  non-residue,  we  notice  first  that  if 

a=±q1q2  •••  qn, 
where  qx,qo,--,qn  are  the  positive  prime  factors  of  a  we  have 

Hence  (a/p)  =  1  or  —  1  according  as  an  even  or  an  odd  number 
of  the  symbols  (±  i/p),  (qi/p), '",  (qn/p)  have  the  value  — 1 ; 
that  is,  a  will  be  a  quadratic  residue  of  all  primes  of  which  an 
even  number  or  none  of  the  factors  ±  i,qXi  '-,qn  are  non-residues. 
To  determine  for  what  values  of  p  the  value  of  (a/p)  is  1,  for 
what  —  1,  it  is  therefore  only  necessary  to  determine  for  what 
values  of  p  the  value  of  each  of  the  symbols  in  the  second  member 
of  1 )  is  +  1,  for  what  —  1.  The  problem  may  be  reduced  there- 
fore to  the  following  three  simpler  ones : 
To  determine 

1.  Of  what  odd  prime  moduli  —  1  is  a  quadratic  residue? 

2.  Of  what  odd  prime  moduli  2  is  a  quadratic  residue  ? 

3.  Of  what  odd  prime  moduli  is  another  positive  odd1  prime 
a  quadratic  residue? 

§  6.    Prime  Moduli  of  which  —  1  is  a  Quadratic  Residue. 

By  trial  —  1  is  seen  to  be  a  residue  of  5,  13,  17,  29  and  a  non- 
residue  of  3,  7,  11,  19,  23,  and  we  are  led  by  induction  to  the  fol- 
lowing theorem: 

1  Primary  prime.     See  p.  193. 


QUADRATIC   RESIDUES.  I  29 

Theorem  3.  The  unit  —  1  is  a  quadratic  residue  of  all  positive 
primes  of  the  form  4n  + 1  and  a  quadratic  non-residue  of  all 
positive  primes  of  the  form  pi  —  i.2 

We  have  (§4) 

^J^m{^lpt  mod  A 

whence,  since  ( —  1 )  (p-X)/2  =  1  or  —  1, 


(¥)-< 


p-i 

i)r- 


Now  p  has  either  the  form  qn  -f-  1  or  4^ —  1,  and  it  is  easily  seen 

that  when  p=4n  + 1,  (—i)<p-v/2  =  i,        aHl^s    4/u*' 

and  when  .    p  =  pi — 1,  (—  t)(p-1>/2  = — ^ 

Therefore  ( — —  J  =  1  when  p  =  \n  +  1 , 


and  I J= — 1  when  p  =  ^n — I. 

Ex.  1.     We  have    ( — 1/13)  =  1    since    13  =  4-3  +  1;   that   is,   the  con- 
gruence x*?= — 1,  mod  13, 

has  roots.     These  roots  are  easily  seen  to  be  5  and  — 5. 
Ex.2.    We    have    ( — 1/23)=  —  1,   since   23  =  4-6 — 1;    that    is,    the 

congruence  x2^ — 1,  mod  23, 

has  no  roots;  a  result  easily  verified. 

§  7.    Determination  of  a  root  of  the  congruence  x2  ==  —  1,  mod 
p,  (p  =  4n  +  1)  by  means  of  Wilson's  Theorem. 

Write  down  the  following  congruences,  which  are  evidently  true : 

2n  +  1  =  —  2n,  mod  p, 

2n  -f-  2  =s  —  {211  —  1 ) ,  mod  p, 

2n  +  3= —  (2n  —  2),  modp, 


4n  =  —  1,  modp, 
2  First  given  by  Fermat ;  first  proved  by  Euler. 
9 


I3O  QUADRATIC   RESIDUES. 

and  the  identical  congruence 

(2ft)  !=  (2^)  !,  mod  p. 
Multiplying  these  congruences  together,  we  obtain 
(4n)!==(— i)2w[(2ft)  !]2,  modp, 

or  (/-  i)|.»J;p=iJ:!Jl  mod/, 

But  by  Wilson's  Theorem 

(p  —  i )  !  s==  —  i,  mod  p, 

whence  {- )  1 1   =  —  i,  mod  py 

and  therefore  *  m  I- )  !,  mod/, 

is  a  root  of  x2== —  I,  mod  p. 

Ex.     By  the  above  theorem  the  congruence 
x2^  —  i,  mod  13, 
has  a  root  x  ==  (    3~     J !  ^=  6 !  ^=  5,  mod  13 ; 

that  is,  52== — 1,  mod  13. 

§  8.    Gauss's  Lemma. 

The  following  theorem  known  as  Gauss's  Lemma,  will  enable 
us  to  determine  (2/p)  and  (q/p). 

Theorem  4.  //  m  be  any  integer  not  divisible  by  p  and  if 
among  the  residues  of  smallest  absolute  value,  mod  p,  of  the 
products  im,  2m,  3m,  ■••,  %(p  —  i)m,  there  be  an  even  number 
of  negative  ones,  m  is  a  quadratic  residue  of  p,  if  an  odd  number, 
m  is  a  quadratic  non-residue ;  that  is,  if  fx  be  the  number  of  nega^ 
tive  residues,  (m/p)  —  ( —  /)**. 

We  shall  illustrate  the  content  of  this  theorem  by  a  numerical 
example. 

Let  />  =  I3  and  w  =  3.  The  residues  of  smallest  absolute 
value,  mod  13,  of  the  integers 

3,  6,  9,  12,  15,  18 

are  3,  6,  —4,-1,  2,  5, 


QUADRATIC   RESIDUES.  I3I 

two  of  which  are  negative.     Hence  3  is  a  residue  of  13 ;  that  is, 

This  is  seen  to  be  true  since  the  congruence 
*2e=3,  mod  13, 

has  the  roots  4  and  —  4. 

To  prove  the  theorem  we  proceed  as  follows : 
Since  m  is  prime  to  p,  the  (p — 1)/2  multiples  of  m 

P—  l  X 

IW,  2m,-'./- m  1) 

2 

are  incongruent  each  to  each,  mod  p.  Their  residues  of  smallest 
absolute  value,  mod  p,  are  therefore  different  integers  of  the 
system 

p  —  1        p  —  3  p  —  3  p  —  I 

2  2  2  2 

Those  which  are  positive  and  belong  therefore  to  the  system 

I,  2,  -,^-— -,  2) 

2 

we  shall  denote  by  blt  b2,  ---,b^.  Those  which  are  negative,  and 
belong  therefore  to  the  system 

_,     _2     ...     -P^l 

1  >  ^1  »  > 

2 

we  shall  denote  by  — ax,  —  a2,--, —  aM. 

Evidently  a^,  a2,  -  •  • ,  a^  belong  to  the  system  2).     Moreover 

2 

We  shall  now  prove  that 

a^,a2,  •■•,aIJi,b1,b2,  '■•,bx 
are  the  integers 

/—  I 


I32  QUADRATIC    RESIDUES. 

in  some  order.  To  do  this  it  will  be  sufficient  to  show  that  no 
two  of  these  integers  are  congruent  to  each  other,  mod  p.  It  is 
evident  that  no  two  a's  are  congruent  to  each  other,  mod  p,  and 
the  same  is  true  of  the  b's.     Also  no  a  is  congruent  to  a  b,  mod  p. 

For  if  di  =  bj,  modp, 

and  if  Km  and  km  be  the  integers  of  1),  of  which  — a-i  and  bj 
are  the  residues  of  smallest  absolute  value,  mod  p,  then 

—  hm  =  km,  modp, 

and  hence  (h  -f-  k)m  =  o,  mod  p, 

which  is  impossible,  for  m  being  prime  to  p,  and  h  and  k  both 
positive  and  <  p/2,  neither  of  the  factors  m  or  h~\-  k  is  divisible 
by  p.     Therefore  the  (p —  i)/2  integers, 

alt  a2,  -••,atL,b1,  b2,  •••,bx, 

are  incongruent  each  to  each,  mod  p,  and  being,  moreover,  all  posi- 
tive and  <  p/2,  must  be  the  integers 

t*St=± 

1,2,         ,  2 

in  some  order. 

Since  — ax,  —  a2,  •••,  —  a  ,blfb2,'-,b\ 

are  residues  of 

/—  l 
itn,  2m,  3«f,  •••  , m,  mod  /, 


we  have 


/—I     ^ 

1 -2.»  -^—  m  f    s(-i)^...^1..^AlmodA 


whence,  since 


/  —  1 


and  this  product  is  prime  to  />,  we  have 

w  2    ■■■(■—  l)*,  mod/. 


QUADRATIC   RESIDUES.  I  33 


But 


(")- 


p-1 

m  2   ,  mod  p, 


and  (-  i)M=  1  or—  1 

Therefore 


(?)-<->• 


We  call  /x  Gauss's  Characteristic. 

§  9.    Prime  Moduli  of  which  2  is  a  Quadratic  Residue. 

We  see  by  any  one  of  the  several  methods  given,  that  2  is  a 

residue  of  the  primes         7,  17, 23, 31, 41, 47, 

which  are  of  the  form  Sn  zb  1,  and  a  non-residue  of  the  primes 

3,  5,  11,  13,  19,  29,  37, 

which  are  of  the  form  8w  ±  3. 

Now  every  odd  prime  is  of  the  one  or  the  other  of  these  forms, 
and  the  truth  of  the  following  theorem  seems  at  once  probable. 

Theorem  4.  The  integer  2  is  a  quadratic  residue  of  all  primes 
of  the  form  8n  ±  1,  and  a  quadratic  non-residue  of  all  primes  of 
the  form  8n±  3.1 

From  Gauss's  Lemma  we  have 


(0- 


where  p  is  an  odd  prime,  and  /*  is  the  number  of  the  integers 

2,4,6,---,/>  —  1,  1) 

whose  residues  of  least  absolute  value,  mod  p,  are  negative.  To 
determine  when  ft  is  even  and  when  odd  we  notice  that  these  fi 
integers  are  those  greater  than  pi 2.  If  we  suppose  the  series  1) 
to  be  formed  by  continued  subtraction  of  2  from  p  —  1  and  write 
it  in  the  form 

P—  I,/>  —  3,"-,p  —  I—  2(fX—  l),p—  I—  2fi,  •••,4,2, 
1  First  given  by  Fermat ;  first  proved  by  Lagrange. 


134  QUADRATIC    RESIDUES. 

we  see  that,  since  there  are  /x  of  its  terms,  beginning  with  p  —  I 
and  going  backwards,  whose  residues  of  least  absolute  value,  mod 
p,  are  negative,  the  smallest  one  of  these  terms  will  be 

P  —  i—  2(/x—  i). 

The  greatest  term  whose  residue  of  least  absolute  value,  mod  p, 
is  positive  is  therefore  p  —  I  —  2ti. 
Hence  we  have 

p-l  -2(/i-  i)>|>^_i_2/v  2) 

From  2)  we  obtain 

4  4 

and  therefore  /*  is  the  greatest  integer  contained  in  the  fraction 
(/>  +  2)/4.     Hence  we  have,  when 

p  =  Sn  ±  i,  fi  =  2n, 

and  when  p  =  Sn  -fc  3,  fi  =  2n  ±  1 ; 

that  is,  fi  is  even  when  p  has  the  form  Sn  ±  1,  and  odd  when  p 

has  the  form  8w  ±  3. 

Hence 

(  -7  J  =  i,  when/>  =  8«  ±  1, 
and 

f  -J==—  1,  when/>  =  8;i:±3, 

and  the  theorem  is  proved. 

We  can  express  this  result  very  conveniently  in  the  following 
manner.     We  observe  that 

£2  j 

when  p  =  Sn  ±  1,  — —  ==  8n2  ±  2n 

Q 

£2  I 

and  when  p  =  Sn  ±  3,  —5 —  =  Sn2  ±6n  +  1 ; 

o 

/>2 j 

that  is,  when  p  =  Sn  ±  1,  ^  is  even, 


QUADRATIC   RESIDUES. 


135 


and  when 


Hence. 


Ex.  1.     We  have 


p  =  8n±  3, 


is  odd. 


GH-"' 


172-1 

1      8      . 


Therefore  2  is  a  quadratic  residue  of  17. 

f        v  112-1 

Ex.2.    We  have     (_)  —  (_!)    8    =(_I)i5__I> 

Therefore  2  is  a  quadratic  non-residue  of  11. 

§  10.    Law  of  Reciprocity  for  Quadratic  Residues. 

It  remains  now  to  answer  the  question :  of  what  odd  primes  is  a 
positive  odd  prime  q  a  residue,  of  what  a  non-residue?  This  is 
answered  by  means  of  a  theorem  which  expresses  the  quadratic 
character  of  q  with  respect  to  p  in  terms  of  the  quadratic  character 
of  p  with  respect  to  q ;  thus  making  the  answer  depend  upon  that 
to  our  first  question,  §  2.  This  theorem,  which  Gauss  has  called 
the  "  Gem  of  the  Higher  Arithmetic,"  is  known  as  the  "  Law  of 
Reciprocity  of  Quadratic  Residues,"  or  more  briefly  as  the 
"  Quadratic  Reciprocity  Law."     It  is  the  following: 

Theorem  5.  Law  of  Reciprocity  of  Quadratic  Residues.1  If 
p  and  q  be  two  different  positive  odd  primes,  the  quadratic  char- 
acter of  q  with  respect  to  p  is  the  same  as  or  different  from  the 
quadratic  character  of  p  with  respect  to  q,  according  as  at  least 
one  of  the  primes  is  of  the  form  4n  -f- 1,  or  both  are  of  the  form 
pi  —  1;  that  is,  if 

p  =  4h  -f- 1  and  q  =  ^k  -f-  1, 
or  ^  =  4/i-|-i  and  g  =  4^ — 1, 

p  =  ^h  —  1  and  q  =  4k-{-i, 


or 


while  if 


p  =  ^h — 1  and  q  =  $k  —  1,       (  — 


(i)  a)- 


1. 


xSee  Bachmann:    Niedere  Zahlentheorie,  pp.    194-318,   for  a  very  full 
discussion  of  this  theorem,  a  list  of  all  proofs  being  given.  . 


I36  QUADRATIC    RESIDUES. 

This  theorem  can  be  expressed  in  a  very  elegant  form,  if  we 
observe  that  the  expression  (p — i)/2-(q  — 1)/2  is  even  when 
one  or  both  of  the  primes  are  of  the  form  4ft  -\-  1,  but  odd  when 
both  are  of  the  form  qn —  1.     We  have,  therefore, 


$(*)-<->?*■ 


qf  \p 

The  proof  which  follows  is  due  to  Pfarrer  Zeller,2  and  depends 
solely  on  Gauss's  Lemma. 
We  have  by  Gauss's  Lemma 


(!)=(-)-. 


J 

where  /a  is  the  number  of  the  products 

\q,2q,--/-^—q,  1) 

whose  residues  of  least  absolute  value,  mod  p,  are  negative  ;  likewise 

where  v  is  the  number  of  the  products 

ip,2p,---,q-^p,  2) 

whose  residues  of  least  absolute  value,  mod  q,  are  negative. 

Hence  (J)^-*-1^ 

The  problem  is  therefore  resolved  into  the  determination  of  those 
cases  in  which  p  -\-  v  is  even  and  those  in  which  it  is  odd.  Denote 
the  residues  of  least  absolute  value,  mod  p,  of  the  products  1)  by 

—  a1}  —  a2,  •••,  —  a^  b1}b2,  •••,  b\» 
and  those  of  the  products  2),  mod  q,  by 

c1?      c2,  •  •  *,      cVy  d1)  d2) ' '  *,  a  p) 
2  Monatsbericht  der  Berliner  Akademie,  December,  1872. 


QUADRATIC   RESIDUES.  I  37 

the  a's,  fr's,  c's  and  d's  all  being  positive.  Since  p  and  q  are  dif- 
ferent from  each  other,  one  must  be  the  greater.  Assume  q  >  p. 
We  divide  now  the  integers  clt  c2,  ',cv,  all  of  which  being  resi- 
dues of  least  absolute  value,  mod  q,  belong  to  the  system 


1,2, 


into  two  classes  according  as  they  are  greater  or  less  than  p/2. 

The  system  of  those  which  are  <  p/2  we  denote  by  Cx  and  the 

system  of  those  >  p/2  by  C2. 

Let  v1  denote  the  number  of  the  integers  Clf  and  v2  that  of  the 

integers  C2. 

The  proof  now  falls  naturally  into  the  following  four  parts : 
i.  That  the  integers,  Clf  are  identical  with  the  b's  and  therefore 

together  with  the  a's  make  up  the  system 

p—i 

whence  /*  +  v  =  - f-  v2. 

ii.  That  the  number,  v2,  of  the  integers  C2  is  odd  or  even 
according  as  the  number  (p-{-q)/4  is  or  is  not  found  among 
them. 

iii.  That  (p  +  q)/4  occurs  among  the  integers  C2,  and  hence 
v2  is  odd,  when  and  only  when  we  have  simultaneously 

p  =  4h — 1  and  q  =  4k  +  I. 
iv.  That  therefore  p-f-  v  is  odd  when  and  only  when  simul- 
taneously p  =  4h  —  1  and  q  =  4k  —  1 . 

The  proof  will  be  rendered  more  intelligible  if  we  consider 
first  the  relation  between  the  four  parts  into  which  we  have 
divided  it. 

Suppose  that  we  have  proved  i,  then 


(I)  a)  -  (-  ■>--. 


I38  QUADRATIC   RESIDUES. 

and  to  prove  our  theorem  it  is  sufficient  to  show  that  (p — i)/2 
+  v2  is  odd  when  and  only  when 

p  =  4h — 1,  q=^4k — I. 

It  is  evident,  however,  that  since  (p — 1)/2  is  even  or  odd 
according  as  p  =  4k  +  1  or  4/t  —  1,  to  show  that  (p  —  1  )/2  +  v2 
is  odd  when  and  only  when  p  =  4h — 1,  q  =  4k — 1,  it  is  suffi- 
cient to  show  that  v2  is  odd  when  and  only  when  p  =  4h — 1, 
q  =  4k-\-i.  Now  the  number  {p  +  q)/4  is  less  than  q/2  and 
greater  than  p/2  and  hence,  */  an  integer,  is  either  one  of  the 
integers  C2  or  one  of  the  d's. 

But  (p  +  q)/4  is  an  integer  only  when  p  =  4h  +  1,  q  —  4k —  1 
or  p  =  4h — 1,  q  =  4k-\-i,  and  hence  can  therefore  evidently 
never  be  among  the  integers  C2  in  the  cases  p  =4/1+  i}q=^k  -4-  1 ; 
and  p=4h —  1,  q  =  4k  —  1.  If  now  we  can  show  that  (p  -\-q)/4 
always  occurs  among  the  integers  C2  when  p=4h  —  1,  q=4k-\-i, 
and  never  when  p=4h-\-if  q  =  4k —  1,  then  to  show  that  v2  is 
odd  when  and  only  when  p  =  4h  —  1,  q  =  4k-{-i,  it  will  be 
sufficient  to  show  that  v2  is  odd  when  and  only  when  (p  +  q)/4 
occurs  among  the  integers  C2.  Therefore  to  show  that  (p —  i)/2 
-\-v2  is  odd  when  and  only  when  />=4/j —  i,  q  =  4k — 1,  it  will 
be  sufficient  to  show  that  (p  +  q)/4  occurs  among  the  integers  C2 
when  and  only  when  p  =  4h —  1,  p  =  4k  -f- 1.  Our  idea  is  there- 
fore to  show  that  the  three  conditions 

p  =  4h—i,  q  =  4k  +  i, 

v2  odd, 

one  of  the  integers  C2, 
4 

are  equivalent,  whence  it  will  follow  that  (p  —  i)/2  +  v2  is  odd 
when  and  only  when  p  =  4h  —  1,  q  =  4k  —  1. 
i.  If  any  integer  of  the  system 

p  —  i 

be  not  an  a  it  must  be  a  b ;  for  as  we  have  already  shown  (Th.  4), 
the  a's  and  b's  together  make  up  this  system.     The  integers  Cx 


QUADRATIC    RESIDUES.  I  39 

belong,  however,  also  to  this  system,  hence  each  of  the  integers 
Cx  must  be  either  an  a  or  a  b.  We  shall  show  that  each  b  is  iden- 
tical with  one  of  the  integers  Cx ;  also  that  no  a  is  identical  with 
any  of  the  integers  Cx  and  hence  the  fr's  and  the  integers  Cx  coin- 
cide. Let  bi  be  any  one  of  the  b's,  and  hiq  that  product  of  the 
system  i)   whose  residue  of  least  absolute  value,  mod  p,  is  bi. 

P 
We  have  then       hiq  =  bi,  mod  p  ;  o  <  hi  <  -; 

that  is,  hiq  =  kip-{-bi,  3) 

where  ki  is  an  integer  such  that 

P 
o<  kip<hiq<-q, 


and  hence 

o<*»<f. 

Therefore  k 

ip  is  one  of  the  products  of  the  system 

2). 

But  from 

3), 

we  have 

kip  =  — bi,  modg, 

where 

P 
o<bi<^. 

Hence  bi  is  one  of  the  integers  Cx. 

But  bi  is  any  one  of  the  b's ;  hence  each  b  is  identical  with  one 
of  the  integers  Cv  Let  now  a}-  be  any  one  of  the  o's  and  hjq  that 
product  of  the  system  1)  whose  residue  of  least  absolute  value, 
mod  p,  is  —  ctj.     We  have  then 

hjq  =  —  dj,  mod  p  ; 

that  is,  hjq  =  kjp  —  a;-,  4) 

where  kj  is  an  integer  >  o  and  <  q/2 ;  for  from  4) 

__hjq  +  aj 
kj~~P       ' 

P  P 

and  hence,  since      o  <  ay  <  -,  and  o  <  hj  <  -, 

—  2 


140  QUADRATIC   RESIDUES. 


we  have 

2*2 

n  ^  hi  ^ 

O  <v  K)  <.                      , 

that  is, 

o<^<*+\ 

which  gives, 

since 

k, 

and  (g  -f  i)/2  are  integers, 

Hence  kjp  is  one  of  the  products  2),  and  since  from  4)  it  follows 

that  kjp==aj,  modg, 

dj  is  a  d  and  therefore  not  one  of  the  integers  Ct,  But  a,  is  any- 
one of  the  a's ;  hence  no  a  can  be  identical  with  one  of  the  integers 
Cv  Hence  the  a's  and  the  integers  Cx  coincide,  and  therefore  the 
a's  and  the  integers  Cx  make  up  the  system 

/-  I 

1,2,         ,         2 

Therefore  fi  -\-  v  = \-  v2. 

ii.  To  prove  now  that  the  number,  v2,  of  the  integers  C2  is 
odd  or  even  according  as  the  number  (p  -j-  q)/4  is  or  is  not  found 
among  them,  let  Ci  be  one  of  the  integers  C2  and 

kip  =  —  a,  modg. 

Here  ki  can  not  be  (q  — 1)/2,  for  we  have 

2  2  2 

that  is, /  ss  - £,  mod  a, 

2  2 

where  (g —  p)/2  is  evidently  positive  and  less  than  a/2,  and  hence 
one  of  the  a"s. 

Therefore  to  each  product,  kip,  of  the  system  2),  whose  residue 
of  least  absolute  value,  mod  q,  taken  positively  is  an  integer  of  C2, 


QUADRATIC    RESIDUES.  I4I 

there  corresponds,  since 

a  product  kjp,  (fcj+  (q  — i)/2),  of  the  same  system,  such  that 

^  2  *' 

We  shall  show  now  that  the  residue  of  least  absolute  value,  mod 
q,  of  kjp,  taken  positively,  is  also  one  of  the  integers  C2. 
Multiplying  5)  by  p,  we  have 

whence  £•/  =  ?  + —  —  kj>, 

or  k.p  m  tilt  -  kj>,  mod  q, 

and  hence  /£•/»  ■  — —  kj>,  mod  q. 

Moreover,  since  kip  =  —  a,  modg, 

we  have  k,p  =  — +  cp  mod  q. 

^  P  Q 

But  since  -  <*/<-, 

2         *      2 

we  have  £<£xf  — *<'£ 

22  2 

Hence ^  is  one  of  the  integers  C2. 

Putting  /+_?  -  ^  =  r.,  6) 

we  see  that  if  kip,  kjp,  be  two  products  of  the  system  2),  such  that 
ki  and  kj  are  connected  by  the  relation  5),  and  if  the  residue  of 
least  absolute  value  of  kip,  mod  q,  be  —  d,  where  C\  is  one  of  the 


142  QUADRATIC   RESIDUES. 

integers  C2,  then  the  residue  of  least  absolute  value  of  kjp,  mod  q, 
is  —  Cj,  where  Cj  is  also  one  of  the  integers  C2. 

Hence  to  each  integer  d  of  C2  there  corresponds  in  this 
manner  another  integer  Cj  of  C2  and  it  is  evident  that  unless  it 
should  happen  that  there  is  one  (or  any  odd  number)  of  these 
pairs  whose  integers  are  identical,  the  number,  v2,  of  the  integers 
C2  will  be  even,  but  if  the  two  integers  composing  each  of  any  odd 
number  of  these  pairs  be  identical,  v2  is  odd. 

If  a  =  Cj,  then  from  6)  it  follows  that 

4       *  4 

Hence  there  is  at  most  one  pair  whose  integers  are  identical  and 
this  case  will  occur  when  and  only  when  (p  +  q)/4  is  one  of  the 
integers  C2.  Hence  v2  is  odd  or  even  according  as  (p-\-q)/4 
does  or  does  not  occur  among  the  integers  C2. 

iii.  To  prove  now  that  (p-\-q)/4  occurs  among  the  integers 
C2,  and  hence  v2  is  odd,  when  and  only  when  we  have  simulta- 
neously p  —  ^h — i,  q  =  4k  +  i, 

we  observe  first  that 

P      P  +  9       g 

2  4  2 

and  hence,  if  (p-\-q)/4  be  an  integer,  it  is  either  one  of  the 
integers  C2  or  a  d. 

In  order  now  that  (p-\-q)/4  may  be  one  of  the  integers  C2 
it  is  necessary  and  sufficient  that  there  shall  be  one,  kp,  of  the 
products  2)  such  that 

/  +  Q 

kp  m  — ,   mod  q  ; 

that  is,  it  is  necessary  and  sufficient  that  there  shall  exist  two 
integers  h  and  k  such  that 

kp-kq-P-±Z,  7) 


QUADRATIC   RESIDUES.  1 43 

and  k  <  -  • 

2 

From  7)  it  follows  that  we  must  have 

(4k  +  i)p=(4h—i)q,  8) 

and  hence  4k  -f-  1  divisible  by  q. 

q 
But  we  have  k  <  -  , 

2 

and  hence  4&  +  1  <  2g. 

Therefore  g  =  4k  +  1, 

and  consequently  from  8)  it  follows  that 

p  =  4h—i; 

that  is,  in  order  that  the  required  integers  h  and  k  may  exist,  p 
and  q  must  have  these  forms.  Moreover,  when  p  and  q  have  these 
forms  the  required  integers  h  and  &  evidently  do  exist. 

Hence  p  =  4h  —  1,  q  =  4k  -\-  1  isa  necessary  and  sufficient  con- 
dition that  (p  +  q)/4  shall  be  one  of  the  integers  C2. 

Therefore  v2  is  odd  when  and  only  when  we  have  simultaneously 

p  =  4h — 1,  and  q  =  4k  +  1. 

iv.  To  prove  now  that  /*  +  v  is  odd  when  and  only  when  we 
have  simultaneously  p  =  4h — 1,  q  =  4k — 1,  we  examine  the 
equation 

j_          P  — T     1 
p-rv=  — r-"2 

and  observe  that 

p  =  4h-\-  1,  q  =  4k  +  1  gives  even,  v2  even,  fi-\-v  even, 

2 

/>  —  1 
p  =  4h  -f- 1,  g  =  4£  —  1  gives  even,  v2  even,  ft  +  v  even, 

p  =  4h — 1,  q  =  4k-{-i  gives  odd,  v2  odd,  /a  -j-  v  even, 

2 

p  =  4J1  —  1  f  g  =  4^  —  1  gives odd,  v2  even,  fi-{-v  odd. 


144  QUADRATIC   RESIDUES. 

Therefore  (| )  (J  )  =  , 

when  at  least  one  of  the  positive  primes  p  and  q  has  the  form 

4W+I,a„d  (f)(J)— * 

when  both  have  the  form  4.11  —  1. 

4  §  11.  Determination  of  the  Value  of  (a/p)  by  means  of  the 
Quadratic  Reciprocity  Law,  a  being  any  Given  Integer  and  p 
a  Prime. 

Before  discussing  the  question  of  what  odd  prime  moduli  is  a 
given  positive  odd  prime  a  quadratic  residue,  which  we  shall  be 
able  to  answer  by  means  of  the  Quadratic  Reciprocity  Law,  we 
shall  illustrate  upon  an  example  how  greatly  the  use  of  this  law 
simplifies  the  determination  of  the  value  of  (a/p),  where  a  and  p 
are  both  given  integers  and  p  an  odd  positive  prime;  that  is,  the 
determination  whether  the  congruence 

„r2==a,  modp, 
has  or  has  not  roots. 
Let  ^  =  365,  mod  1847,  0 

be  the  congruence  under  discussion,  1847  being  a  prime.1 
We  have 


V1847/     V1847/  V1847/ 


847/       \i847/  \i847> 
Then  since  5  is  a  prime  of  the  form  4^+1,  we  have 

§47N 


V1847/      v 


847/       V    5 
and  since  1847  =  2,  mod  5, 

5  being  of  the  form  8n  —  3. 

Hence  I  — — 

\1847 

1  Dirichlet-Dedekind :  p.  103. 


QUADRATIC   RESIDUES.  1 45 

Likewise  since  73  is  of  the  form  411  +  1, 
and  1847  =  22,  mod  73, 

we  have 

(is^j =  V73-)  =  (jj)  =  \y3 )  xjj)  ■ 

But  (£)-,; 

since  73  is  of  the  form  8n  -\-  1,  and  therefore 

VT847/ =  (73/ ' 

Again  since  73  is  of  the  form  4»  +  *  and  73  =  7,  mod  11, 

(M)-(H)-(f,)- 

Since  7  and  11  are  both  of  the  form  471 —  1, 

(f.) =-(")-  ©-©©—■ 

Therefore  (^)  =  (-1)  (_i)  =  i ; 

that  is,  1 )  is  solvable. 

Its  roots  can  be  shown  to  be  ±  496. 

§  12.  Determination  of  the  Odd  Prime  Moduli  of  which  a 
Given  Positive  Odd  Prime  is  a  Quadratic  Residue. 

Let  q  be  an  odd  positive  prime. 

We  are  to  determine  for  what  positive  odd  prime  values  of  p 
the  value  of  (q/p)  is  1,  for  what  —  1. 

By  means  of  the  Quadratic  Reciprocity  Law  we  are  able  to 
make  the  solution  of  this  problem  depend  on  that  of  the  simpler 
one,  which  we  have  already  solved ;  that  is,  the  division  of  all 
rational  integers  into  two  classes,  one  of  which  contains  all  resi- 
dues of  q  and  the  other  all  non-residues. 

Let  rx,r2,  '-,rt  and  nx,n2,  ••-,«*  be  respectively  the  incongruent 
quadratic  residues  and  non-residues  of  q.  Then  an  integer  is  a 
10 


I46  QUADRATIC   RESIDUES. 

residue  or  non-residue  of  q  according  as  it  is  contained  in  one  of 
the  forms  rt  +  kq,  r2-\-kq,-",rt-\-kq,  1 ) 

or  in  one  of  the  forms 

n1  +  kq,n2  +  kq,--',nt  +  kq.  2) 

It  is  necessary  now  to  distinguish  two  cases  according  as  q  has 
the  form  411  +  1  or  4^ —  1. 
i.  g  =411+1. 


(*)-(& 


Then 

ft        \q 

that  is,  q  is  a  quadratic  residue  or  non-residue  of  p  according  as  p 
is  a  quadratic  residue  or  non-residue  of  q.  Hence  q  is  a  residue 
of  all  positive  odd  primes  contained  in  the  forms  1)  and  a  non- 
residue  of  all  positive  odd  primes  contained  in  the  forms  2). 

Ex.    Let  q  =  13. 

The  residues  of   13  are  1,  3,  4,  9,   10  and   12,  the  non-residues  2,   5, 
6,  7,  8  and  11. 

Hence  13  is  a  residue  of  all  primes  of  the  forms 

1  +*I3&,  3  +  I*3*j  4  +  13^  9  +  13^ Jto  +  13&,  12  +  13k, 
and  a  non-residue  of  all  primes  of  the  forms 

2  +  izk,  5  +  izk,  6  +  136,  7  +  13^,  8  +  13k  11  +  13^ 
ii.  q  =  4.n —  1. 

We  must  further  divide  this  case  into  two  parts  according  as  p 
has  the  form  4m  +  1  or  4m —  1. 

a)  p  =  4tn-\-i. 


&  -  (£) 


Then 

pi        \q 

and  q  is  seen  to  be  a  quadratic  residue  of  all  primes  of  the  form 
4m  +  1  contained  in  the  forms  1 )  and  a  non-residue  of  all  primes 
of  the  form  4m  +  1  contained  in  the  forms  2). 

b)  p  =  4m — 1. 

(JHf)    • 


QUADRATIC   RESIDUES.  1 47 

and  q  is  seen  to  be  a  quadratic  residue  of  all  positive  primes  of 
the  form  4m —  1  contained  in  the  forms  2)  and  a  quadratic  non- 
residue  of  all  positive  primes  of  the  form  4m  —  1  contained  in 
the  forms  1). 

The  primes  p  are  in  this  case  seen  to  be  subjected  to  two  con- 
ditions, first  that  they  shall  give  with  respect  to  the  modulus  4  the 
residues  1  or  —  1,  and  secondly  with  respect  to  modulus  q  the 
residues  rlyr2,  --,rt  or  n19 n2,  •  •  • , nt . 

By  Chap.  Ill,  §  14,  we  can  find  the  forms  which  the  numbers 
must  have  in  order  to  satisfy  both  of  these  conditions. 

Ex.1     Let  q  =  19. 

The  residues  of  19  are 

i,  4,  5,  6,  7,  9,  11,  16  and  17, 
and  the  non-residues 

2,  3,  8,  10,  12,  13,  14,  15  and  18. 

Hence  19  is  a  residue  of  all  positive  primes  of  the  form  4m  -\-  1  con- 
tained in  the  forms 

19*  +  i,  19* +  4,  I9&  +  5,  19^  +  6,  19^  +  7/ 

19*4-9,  19*  + n,  19*  + 16,  19*  4"  17,  3) 

and  of  all  positive  primes  of  the  form  \m  —  1  contained  in  the  forms 
19*  4"  2>  19^  4"3?  I9#  +  8,  19*  4~  IO>  J9^  4~  I2, 

19*  +  13,  19*  4-14,  19&+  15,  I9& 4- 18.  4) 

On  the  other  hand  19  is  a  non-residue  of  all  positive  primes  of  the 
form  4W  —  1  contained  in  the  forms  3)  and  of  all  positive  primes  of 
the  form  4m  -f-  1  contained  in  the  forms  4).  By  Chap.  Ill,  §14,  we  may 
combine  the  two  conditions  thus  imposed  upon  p  into  a  single  one  and 
say  that  19  is  a  quadratic  residue  of  all  primes  of  the  forms 

76*4-1,  3,   5,   9,   15,    17,   25,  27,   31,  45,   49,   51,   59,   61,   67,   71,   73,   75, 

and  a  quadratic  non-residue  of  all  primes  of  the  forms, 

76*4-7,  11,  13,  21,  23,  29,  33,  35,  37,  39,  41,  43,  47,  53,  55,  63,  65,  69. 

§  13.  Determination  of  the  Odd  Prime  Moduli  of  which  any 
Given  Integer  is  a  Quadratic  Residue. 

It  was  shown  in  §  10  that  the  solution  of  this  problem  could  be 
made  to  depend  upon  the  solution  of  the  three  simpler  problems, 
to  determine : 

1  Wertheim :  p.  220. 


I48  QUADRATIC   RESIDUES. 

i.  Of  what  odd  prime  moduli  —  1  is  a  quadratic  residue. 

ii.  Of  what  odd  prime  moduli  2  is  a  quadratic  residue. 

iii.  Of  what  odd  prime  moduli  another  positive  odd  prime  is 
a  quadratic  residue. 

These  problems  have  all  been  solved  and  we  are  now  prepared 
to  solve  the  general  question  proposed  originally  in  §2;- that  is, 
to  determine^  for  what  _o^_j^ime~^aiues  of  p  the  value  ofJj^Q^, 
is  i  and  for  what  —  1,  a  being  any  given  integer.  Assuming  that  . 
a  contains  no  square  factor  and  by  pi  denoting  —  1  or  any  positive 
prime  factor  of  a,  we  have  for.  each  pi  two  systems  of  forms,  one 
of  which  contains  all  positive  odd  primes  of  which  pi  is  a  residue, 
the  other  all  positive  odd  primes  of  which  pi  is  a  non-residue. 

The  positive  odd  primes  of  which  a  is  a  residue  will  be  those 
which  are  contained  in  none  or  an  even  number  of  the  second  set 
of  forms.  Having  obtained  for  each  pi  these  two  systems  of 
forms  the  solution  of  the  problem  reduces  to  that  of  finding  an 
integer  which  gives  certain  residues  with  respect  to  ^ach  one  of 
a  series  of  moduli  (Chap.  Ill,  §14).  A  single  example  must 
suffice  here  to  illustrate  the  application  of  this  method.  For  an 
extended  discussion  of  it  with  numerous  examples  see  Wertheim, 
pp.  221,  and  for  the  solution  of  this  problem  as  well  as  the  more 
general  one,  where  the  modulus  is 'also  composite,  see  Dirichlet- 
Dedekind,  Bachmann  and  Mathews,  where  by  an  extension  of 
Legendre's  symbol  a  simplification  is  effected. 

Ex.    Let  a  =  —  15. 

(^)=(t)(I)0) 

Two  cases  must  now  be  distinguished  according  as  p  has  the  form 
4fei  +  1  or  4k,  +  3. 

If  /,  =  4*1  +  1,  (t?)'** 

and  (-  )  =  ( --  )  =       1  when  />  =  ^2-^1^ 

and  as  —  1  when  p  as  3&?  -f  2 

If  /,^4^  +  3,  ^)=:_I, 


QUADRATIC   RESIDUES.  1 49 

and  [j\  =-  (0  =      I  when  p  =  3k2  +  2, 

and  rs  —  1  when  p  =  $k2  -f  1. 

In  both  cases 


©-(3 


1  when  p  =  5&3  +  X  or  5&3  +  4, 

and  =  —  1  when  />  =  5&3  +  2  or  5^3  +  3. 

In  order  now  that  —  15  shall  be  a  residue  of  p,  p  must  have  such  a 
form  that  either  none  or  two  of  the  symbols  ( — i/p),  (3/p),  (5//O 
have  the  value  —  1. 

Hence  — 15  is  a  residue  of  all  primes  which  are  contained  simulta- 
neously in  the  forms  of  one  of  the  following  sets : 

4&i  +  i,  3&2  -f  1,  5^3  +  1,  which  give  p  =  60k  +    1,  1 ) 

4&i  +  i,  3&2  +  1,  5^3  +  4,  which  give  p  =  60k  -f  49,  2) 

4&1  +  1,  3&2  +  2,  5^3  +  2,  which  give  p  =  60k  +  17,  3,) 

4^1+1,  3^2  +  2,  5&3  +  3,  which  give  />  =  60k  +  53,  4.)- 

4&i  +  3,  3&  +  1,  5^3  +  1,  which  give  p  =  60k  -f-  31,  5) 

4&i  +  3,  3&2  +  1,  5^3  +  4,  which  give  p  =  60k  +  19,  6) 

4&i  "k  3,  3k  +.2,  5^3  +  2,  which  give  />  =  6ofc  +  47,  7)  ' 

4&i  +  3,  3^2  +  2,  5^3  +  3,  which  give  p  =  60k  +  23.  8)- 

■    V 
We  can  easily  combine  1)  and  5),  8)  and  6),  3)  and  7),  4)  and  8),  and 

♦obtain  as  the  forms  of  the  positive  odd  primes  of  which  —  15  is  a  residue 

30&  +  1,  17,  19,  23. 

Similarly   we  find  that  — 15   is   a   non-residue   of   all   positive  primes 

contained  in  the  forms 

3o£  +  7,   n,    13,  29. 
j. 

§  14.    Other  Applications  of  the  Quadratic  Reciprocity  Law. 

We  shall  now  give  a  few  theorems  in  the  proof  of  which  the 
Quadratic  Reciprocity  Law  and  its  two  subsidiary  theorems  will 
be  found  useful. 

Theorem  6.  There  are  an  infinite  number  of  positive  primes 
of  each  of  the  forms  4n-\-  i  and  4n  —  i.1 

Observing  that  every  prime  is  of  one  of  these  forms,  we  pro- 

1  See  Chap.  II,  §  6. 


I50  QUADRATIC    RESIDUES. 

ceed  to  prove  that  there  is  an  infinite  number  of  primes  of  the 
form  4n  -f-  1. 

Suppose  that  there  is  only  a  finite  number  of  positive  primes 
Pi>  p2> ' '  '■■>  Ps,  of  the  form  4.W  +  1.     Form  the  integer 

(2p1p2--p8y  +  i=a, 

which  is  of  the  form  4%  -j-  1. 

It  is  divisible  by  no  prime  q  of  the  form  4% — 1,  for,  if  this 
were  the  case,  we  should  have 

{2pxp2---  psy  =  —  1,  modg; 

that  is,  —  1  would  be  a  quadratic  residue  of  q  which  is  impossible 
because  q  is  of  the  form  471 —  1. 

Moreover,  a  is  not  divisible  by  any  of  the  primes  2,  plt  p2,  •••,/>«. 
Hence  a  is  itself  a  prime  of  the  form  4n-\-  1,  different  from  each 
of  the  primes  pltp2,  •••,^,  or  is  a  product  of  such  primes.  But 
this  is  contrary  to  our  assumption  that  there  are  no  primes  of  the 
form  4M+  1  other  than  plt  p2,  ••*,£«.  Therefore  the  number'  of 
positive  primes  of  the  form  471  — |—  1  is  infinite. 

To  prove  now  that  there  is  an  infinite  number  of  positive  primes 
of  the  form  4%  —  1,  we  assume  as  before  the  contrary  to  be  true; 
that  is,  that  there  are  only  a  finite  number  of  positive  primes 
Qi>  <?2>  "'fit  °f  tne  form  4%  —  1,  qt  being  the  greatest. 

Form  the  integer  zq1q2  •  •  •  qt  +  1  =  b. 

It  is  greater  than  qt  and  is  not  divisible  by  any  of  the  primes 
2>  Qi>  #2>  "'i9.t*  Hence,  if  it  be  not  prime,  its  prime  factors  must 
all  be  of  the  form  411  -{-  1. 

Let  2q1q2'--qt  +  i=p1p2'"pr9  1) 

where  px  ==  1 


/>2  =  i 


mod  4. 


prwmi  ■ 
Multiplying  these  congruences  together,  we  have 

•  PiP2'"pr=i,  mod 4, 


QUADRATIC   RESIDUES.  I  5  I 

whence  2^xg2  •  •  •  qt  +  i  =  i,  mod  4, 

and  hence  QiQ2'"Qt         =  0,  mod 2.  2) 

But  2)  is  impossible  since  q1,q2,'",qt  are  all  primes  of  the 
form  4n —  1. 

Hence  1 )  is  impossible  and  b  is  either  itself  a  prime  of  the  form 
4n  —  1  or  is  a  product  of  primes  of  this  form,  all  of  which  are 
greater  than  qt.  Therefore  the  number  of  positive  primes  of  the 
form  4-n  —  1  is  infinite. 

Theorem  7.  Every  prime  of  the  form  22  +  1  has  a  primitive 
root. 3. 

■  Let  />  =  22"  +1. 

If  3  be  a  primitive  root  of  p,  then  each  of  the  (p  —  2)  powers  of  3 

must  be  incongruent  to  1,  mod  p.  tt) 

If,  however,  3*===  1,  mod  p,  where  o<t<p  —  1,  p  being  positive, 
then,  by  Chap.  Ill,  Th.  25,  it  follows  that 

P  —  iso,  mod/,      ?>i.ji_ 

and,  since  p  —  1  =  22" , 

/  —  2W 

l  —  4    , 

and  the  greatest  possible  value  of  t  will  be  22n  _1.  In  order,  there- 
fore, that  3  may  be  a  primitive  root  of  p,  it  is  necessary  and  suffi- 
cient that  the  following  2n  —  1  incongruences  should  hold 

3      *i, 

322    4u,  },  mod  p. 

32       *ii 

A  sufficient  condition  for  this  is  that  the  last  of  these  incon- 
gruences should  hold,  for  if  any  one  of  the  previous  ones  did  not 
hold,  all  following  ones  would  not  hold. 
We  have  therefore  only  to  prove 

^~l^i,  mod/>; 


I52  QUADRATIC    RESIDUES. 

that  is  32  ^1,  mod/>.  3) 

But  when  3)  is  satisfied,  3  is  a  quadratic  non-residue  of  p,  and 

vice  versa.     Hence  we  have  only  to  prove  (3//O  = —  I. 
Since  p  is  of  the  form  4^+1,  we  have 

/" 

W  4) 


(!)-(!) 


But  2  = — I,  mod  3, 

whence  22"  =  ( —  1 ) 2"  ■■  1 ,  mod  3. 

Therefore  22n  +  1  =  2,  mod  3, 

whence  from  4)  it  follows  that 


©- 


Therefore  3  is  a  primitive  root  of  every  prime  of  the  form 

22"  +  I. 

The  theorem  just  proved  bears  an  interesting  relation  to  the 
problem  of  the  construction  of  regular  polygons  of  a  prime  num- 
ber of  sides  with  ruler  and  compasses ;  the  construction  is  possible 
only  when  p  is  a  prime  of  the  form  22"  -f-  1,  and  can  be  accom- 
plished by  means  of  a  primitive  root  of  p.1 

Theorem  8.  Every  positive  prime  p  of  the  form  4q  -f- 1,  where 
q  is  a  positive  prime,  has  2  as  a  primitive  root. 

If  2  be  a  primitive  root  of  p,  then  each  of  the  p  —  2  powers  of  2 

2,22,  ..-,2P-2 

must  be  incongruent  to  1,  mod  p. 

If,  however,  2  appertains  to  an  exponent  t,  mod  p,  less  than 

p — 1,  then  2*e=i,  mod/>,  5) 

1  See  Klein :  Ausgewahlte  Fragen  der  Elementar  Geometrie,  p.  13. 
Gauss:  Disq.  Arith.,  Sect.  Sept.  Works,  Vol.  I,  p.  412.  Bachmann : 
Die  Lehre  von  der  Kreisteilung,  p.  57  and  Vor.  7th. 


QUADRATIC   RESIDUES.  I  53 

and  by  Chap.  Ill,  Th.  25, 

p  —  1=0,  mod  t, 

whence  4q  mm  o,  mod  /. 

Hence,  since  q  is  a  prime,  we  can  have  as  possible  values  of  t  only 
2, 4,  q  or  2q. 

It  is  necessary  and  sufficient  to  show  that 

24=)si,  mod/>,  and  22«=4=i,  mod/>, 

for,  if  22a=i,  mod  p,  then  24  a»l,  mod/>, 

and,  if  2«s=i,  mod  />,  then  22«e=  i,  mod/>. 

To  prove  24  4a  1,  mod  /> ; 

that  is,  15^0,  mod/>, 

it  is  sufficient  to  notice  that  the  only  primes  which  divide  15  are 
3  and  5,  neither  of  which  is  of  the  form  4*7  -+-  1,  when  q  is  a  prime. 

Hence  24^i,  mod/>. 

To  prove  22«^  1,  mod  /> ; 

that  is,  2(*>-1)/2  4=  1,  mod  p, 

we  need  only  show  that 

(7)— 

we  have  I— J  =  (-  1)   8   =  (-  l)^*-  -  1, 

for  if  ^  =  2,  />,  =4*7  -j-  1,  is  not  a  prime  and  therefore  q  is  always 
odd,  whence  it  is  evident  that  2q2  +  g  is  an  uneven  integer. 

Hence  22«4S  l>  mod/>. 

Therefore  5)  holds  for  no  value  of  t  less  than  p  —  1. 

Hence  2  is  a  primitive  root  of  every  positive  prime  of  the  form 
4q  -\-  1  when  q  is  a  positive  prime. 

Examples. 

1.  Determine  the  prime  moduli  of   which  30  is   a  quadratic 
residue. 


I  54  QUADRATIC    RESIDUES. 

2.  Has  the  congruence 

jtr2  =  H35,  mod  231 1, 
roots  ? 

3.  Solve  the  congruences  :1 

a)  $x2 —  &r —   3  =  0,  mod  23.  x  =  8  or  12,  mod  23. 

b)  3*^  +  4*  +    5  =  0,  mod  20.  .r  =  — 3,  — 5,  7,  5,  mod  20. 

c)  Jx2 —  3;tr-(- 11  =0,  mod  19.  ;fe=5,  9,  mod  19. 

d)  5a-2  —  $x —   2  =  0,  mod  12.  x  =  —  2, 1,2,  5,  mod  12. 

e)  3jf2  +  4^+    9  =  0,  mod  12.  arss — 3,  3,  mod  12. 
/)  Zx%  +    x —   4  — °>  mod  10.  x  =  1,2, 6,7,  mod  10. 

4.  Show  that  among  the  numbers  of  a  reduced  residue  system, 
mod  pn,  where  p  is  a  prime  different  from  2,  there  are  exactly  as 
many  quadratic  residues  as  non-residues  of  pn.2 

5.  Show  that  every  quadratic  residue  of  p  is  also  a  quadratic 
residue  of  pn,  and  that  every  non-residue  of  p  is  also  a  non- 
residue  of  pn.s 

6.  The  numbers  a  and  p  —  a,  where  p  is  a  prime,  have  the  same 
or  opposite  quadratic  characters,  mod  p,  according  as  p  is  of  the 
form  4W  -}- 1  or  4n  —  1 . 

1  Wertheim :  Anfangsgriinde  der  Zahlenlehre,  1902,  pp.  320-322.  This 
book  contains  many  numerical  examples  and  should  be  consulted  by  every 
one  interested  in  such  work.  It  also  contains  many  interesting  historical 
notes  and  some  useful  tables,  and  is  in  many  ways  a  good  book  for  a 
beginner  to  read. 

2  Gauss :  Disq.  Arith.,  Art.  100 ;  Works,  Vol.  I. 

3  Ibid.,  Art.  101. 


CHAPTER   V. 
The  Realm  k(i).1 

§  i.    Numbers  of  k(i).    Conjugate  and  Norm  of  a  Number. 

The  number  V —  I,  that  we  shall  as  usual  denote  by  i,  is  defined 

by  the  equation  x-  +  I  =  o  I ) 

which  it  satisfies. 

Every  number  of  k(i)  is  a  rational  function  of  i  with  rational 
coefficients  (Chap.  I,  §3),  and  since  by  means  of  the  relation 
i2  =  —  1  the  degree  of  any  rational  function  of  i  may  be  reduced 
so  as  to  be  not  higher  than  the  first,  every  number,  a,  of  k(i) 
has  the  form 

a  =  a1  +  b1i 
a  2  -f  b2i ' 

where  alt  bly  a2,  b2  are  rational  numbers,  or,  multiplying  the  numer- 
ator and  denominator  of  this  fraction  by  a2  —  b2i,  we  have 

axa2  -f-  bxb2      a2bx  —  aj>2  . 
<*2  ■+-  b2  ai  +  b,2 

that  is,  every  number,  a,  of  k(i)  has  the  form 

a  =  a  -{-  hi, 

where  a  and  b  are  rational  numbers.'' 

The  other  root  — i  of  the  equation  1)  defines  the  realm  k{ — i) 
conjugate  to  k{%)  (Chap.  I,  §  4).     These  two  realms  are  identical, 

1  Gauss  :  Th.  Res.  Biquad.  Com.  Sec,  Works,  Vol.  2,  p.  95,  f.  f.  Dirichlet- 
Dedekind :  §  139.  Weber :  Algebra,  Vol.  I,  §  173.  Dedekind :  Sur  la 
theorie  des  nombres  entiers  algebrfuques ;  Bulletin  des  Sc.  Math.,  1st  Ser., 
Vol.  XI,  and  2d  Ser.,  Vol.  I.  Bachmann :  Die  Lehre  von  der  Kreisteilung, 
12th  Vor.     Cahen:  pp.  354-367. 

2  Throughout  the  remainder  of  this  book  letters  of  the  Latin  alphabet 
will  always  denote  rational  numbers  (except  in  £(0,  where  *  =  V  —  1) 
while  letters  of  the  Greek  alphabet  will  denote  the  general  numbers  of 
the  realm  under  discussion,  which  may  or  may  not  be  rational  numbers. 

155 


I56  THE   REALM    k(i). 

• 

for  i  is  a  number  of  k( — i)  and  — i  is  a  number  of  k{i)  (Chap.  I, 
§3).  The  number  a  —  bi,  obtained  by  putting  — i  for  i  in  any 
number  a,  =a  -f-  bi,  of  k(i),  is  the  conjugate  of  a  and  is  denoted 
by  a';  for  example,  3  +  2.1  and  3  —  2,i  are  conjugate  numbers 
(Chap.  I,  §4). 

A  rational  number  considered  as  a  number  of  k(i)  is  evidently 
its  own  conjugate. 

It  is  easily  seen  that  the  conjugate  of  a  product  of  two  or  more 
numbers  of  k(i)  is  equal  to  the  product  of  the  conjugates  of  its 
factors;  that  is,  if  fx  =  ap,  then  /i=a,p'.  The  product  of  any 
number,  a,  of  k(i)  by  its  conjugate  is  called  the  norm  of  a  and  is 
denoted  by  n[a]  ;  that  is, 

n[a  +  bi]  =  (a  +  bi)  (a—bi)  =a2  +  b2 
For  example: 

*(3  +  2i]=(s  +  2i)  (3  —  21)  =  13, 
and  n[s]==5'5  =  25- 

We  observe  that  the  norms  of  all  numbers  of  k(i)  are  positive 
rational  numbers. 

Theorem  i.  The  norm  of  a  product  is  equal  to  the  product  of 
the  norms  of  its  factors;  that  is,  n[a/3]  =n  [a]  -n[/3]. 

For  n[ap]=ap-a'F 

=  «  [a]  •«[/?]. 

Every  number,  a,  of  k(i)  satisfies  a  rational  equation  whose 
degree  is  the  same  as  that  of  the  realm,  that  is,  the  second,  and 
whose  remaining  root  is  the  conjugate  of  a,  for  the  equation 
having  for  its  roots  a,=a-\-  bi,  and  a',  =a — bi,  where  a  and  b 
are  rational  numbers,  is 

x2  —  2ax  +  a2-\-b2  =  o;  2) 

and  this  is  of  the  form 

x2  +  px  +  q  =  o,  3) 

where  p  and  q  are  rational  numbers. 


THE   REALM    k(i).  I  57 

If  b=o,  that  is,  if  a  =  a',  the  equation  2)  is  reducible,  becoming 
O  —  a)2  =  o, 
and  the  rational  equation  of  lowest  degree  that  a  satisfies  is 

x  —  a  =  o 

If  ,&=j=o,  that  is,  if  a  =4=  a',  the  equation  2)  is  irreducible,  and 
hence  is  the  single  rational  equation  of  lowest  degree  and  of  the 
form  3)  satisfied  by  a  (Chap.  I,  §  2). 

We  observe,  therefore,  that  the  numbers  of  k(i)  fall  into  two 
classes  according  as  the  irreducible  equations  of  lowest  degree 
satisfied  by  them  are  of  the  first  or  second  degree.  Those  of  the 
second  class,  that  is,  those  which  satisfy  irreducible  rational  equa- 
tions of  the  same  degree  as  that  of  the  realm,  are  called  primitive 
numbers  of  k(i). 

The  numbers  of  the  first  class,  that  is,  those  which  satisfy  irre- 
ducible rational  equations  of  a  degree  lower  than  that  of  the  realm, 
are  called  imprimitive  numbers  of  k(i). 

The  imprimitive  numbers  of  k(i)  are  evidently  the  rational 
numbers. 

All  numbers  of  the  realm  R  being  included  among  those  of  the 
realm  k(i),  R  is  said  to  be  a  sub-realm  of  k(i).  It  is  easily  seen 
that  k(i)  may  be  defined  by  any  one  of  its  primitive  numbers,  but 
by  none  of  its  imprimitive  numbers. 

The  constant  term  of  the  rational  equation  of  the  form  3)  whose 
roots  are  a  and  a'  is  seen  to  be  n[a]. 

In  general,  each  number  a,  of  a  realm,  k(&),  of  the  nth.  degree  satisfies 
a  rational  equation  whose  degree  is  the  same  as  that  of  the  realm  and 
whose  remaining  roots  are  the  n — 1  conjugates  of  a  (see  Chap.  VIII, 
Th.  4). 

§2.    Integers  of  k(i). 

To  ascertain  what  numbers  of  k(i)  are  algebraic  integers  we 
may  consider  separately  the  two  classes  of  numbers  of  the  realm, 
the  imprimitive  numbers  being  at  once  disposed  of  by  remember- 
ing that  a  rational  number  is  an  algebraic  integer  when  and  only 
when  it  is  a  rational  integer. 


I58  THE   REALM    k(i). 

To  determine  when  a  primitive  number  a  is  an  algebraic 
integer,  we  observe  that  the  necessary  and  sufficient  condition  that 
a  shall  be  an  algebraic  integer  is  that  the  coefficients  of  the  single 
rational  equation  of  lowest  degree, 

x2  +  px  +  q  =  o, 

satisfied  by  a  shall  be  integers  (Chap.  II,  Th.  4). 

But  —  p  =  a-\-a',  and  q  =  aa' 

and  hence  the  necessary  and  sufficient  conditions  that  a  shall  be 
an  algebraic  integer  are  that  a  +  a!  and  aa'  shall  be  rational 
integers.2 

If  we  write  a  in  the  form  a  -\-  bi,  where  a =  ax/cx,  and  b  =  bx/cx, 
#!,  bx,  cx  being  rational  integers  with  no  common  factor,  these 
conditions  become 

-J— - — L    -f —  =  —  r=a  rational  integer,  1) 

c1  cx  c± 

(  °±±M  )  (  ^M  )  .  k^l  =  a  rational  integer.      2) 

One  at  least  of  the  three  following  cases  must  occur: 
i.  q4=2  or  1;       ii.  c1  =  2;       iii.  ^=1. 

We  shall  show  that  i  and  ii  are  impossible. 

i.  If  c1=%=2  or  1,  then  by  virtue  of  1)  ox  and  c1  would  have  a 
common  factor  that  by  virtue  of  2)  would  be  contained  in  bx  also. 
But  this  is  contrary  to  our  hypothesis  that  alt  blt  cx  have  no  com- 
mon factor.    Hence  i  is  impossible. 

ii.  If  cx  =  2,  then  by  virtue  of  2)  ax2  +  bx2  would  be  divisible 
by  22  and  hence  ax  and  bx  each  by  2 ;  that  is,  ax,  bx,  cx  would  have 
the  common  factor  2,  which  is  contrary  to  our  hypothesis.  Hence 
ii  is  impossible. 

Hence  cx  =  1 ;  that  is,  a  and  b  are  rational  integers. 

2  This  is  a  special  case  of  the  general  theorem  that  a  necessary  and 
sufficient  condition  for  an  algebraic  number  a  to  be  an  integer  is  that 
all  the  elementary  symmetric  functions  of  a  and  its  conjugates  shall  be 
rational  integers. 


THE   REALM    k(i).  I  59 

Thus  all  integers1  of  k(i)  have  the  form  a  +  bi,  where  a  and  b 
are  rational  integers,  and  all  numbers  of  this  form  are  integers  of 
k(i).  If  b  =  o,  we  obtain  the  rational  integers.  The  conjugate 
of  any  integer  of  k(i)  is  evidently  also  an  integer,  and  the  norm 
of  any  integer  of  k(i)  is  a  positive  rational  integer.  We  observe 
that  in  k(i),  as  in  R,  the  sum,  difference  and  product  of  any  two 
integers  are  integers.2 

§3.    Basis  of  k(i). 

Any  two  integers  a>lf  o>2  of  k(i)  are  said  to  form  a  basis  of  the 
realm  if  every  integer  of  the  realm  can  be  represented  in  the 
form  a1o)1  +  a2w2,  where  alt  a2  are  rational  integers.3 

It  is  evident  that  all  numbers  of  the  form  a^  -f-  a2w2  are  in- 
tegers of  k(i).  We  have  already  seen  that  I  and  i  form  a  basis 
of  k(i)  ;  that  they  are  not  the  only  integers  of  k{i)  having  this 
property  is  easily  shown. 

For  example:  1  +  *,  3  +  2/  is  also  a  basis;  for  if  a-\-bi  be  any  integer 
of  k(i),  then  from 

a  +  bi  =  (h(i  -ft)  +02(3  +  20, 
we  have  ai  +  3a*  =  a, 

fli  +  2a2  =  b, 
giving  ai  =  —  2.a  +  3b, 

Oz  =  a  —  b, 

which  are  rational  integers  since  a  and  b  are  rational  integers. 
We  have 

a  + &*•=(- 2a +  3fc)  (1 +0 +  (<*-&)  (3 +  2*')- 

1  Throughout  the  discussion  of  k(i)  the  term  integer  wiH  be  used  to 
denote  any  integer  of  the  realm  either  complex  or  rational. 

2  It  is  true,  in  general,  that  the  sum,  difference,  and  product  of  any 
two  algebraic  integers  is  an  algebraic  integer  (see  chap.  IX,  Th.  8,  Cor.  2). 

3  There  exist  in  every  realm  of  the  nth  degree  n  integers  «i,  w2,  •••,  <»n, 
such  that  every  integer  0  of  the  realm  has  the  form 

6  =  axWi  +  a&z  +  •  •  •  +  (hio>n, 

where  au  a2,  •••,  an  are  rational  integers.  In  the  definition  here  given  I 
have  followed  Hilbert  (see  H.  B.,  §4).  The  basis  defined  above  is  some- 
times called  a  minimal  basis  of  the  realm  (see  Weber:  Algebra,  Vol.  II, 
§H5). 


i6o 


THE   REALM    k(i). 


For  example ;  8  +  5*  =  —  ( i  + 1)  +  3  (3  +  2*) . 

Every  integer  of  the  realm  is  therefore  expressible  in  the  form 

ai(i+0  +  *(3+*0» 
where  at  and  02  are  rational  integers. 
Hence  1  +  i,  2>-\-2i  is  a  basis. 
We  observe  that  the  determinant  of  the  coefficients1  of  1  +  *  and  3  +  2%  is 

1     1 

3     2 
this  being  a  particular  case  of  the  following  theorem. 

Theorem  2.     //  mxt  o>2  fo  a  basis  of  k(i),  the  necessary  and 
sufficient  condition  that 


1) 


where  alt  a2,  blt  b2  are  rational  integers,  shall  be  also  a  basis  of 
k(i)  is 

at    a. 


bx     b2 


±  1.  2) 

This  condition  is  necessary;  for,  if  c^*, o>2*  be  a  basis,  we  have 


,2  =  &1*»1*  +  &2*os 


3) 


4) 
5) 


where  fl^*,  a2*,  &x*,  b2*  are  rational  integers,  and  substituting  the 
values  of  c^*,  w2*  from  1)  in  3),  we  have 

g>i=  (fl^*^  -f-  02*^i)wi  +  (ai*°%  +  a2*b2)<»2> 
From  4)  and  5)  it  follows  that 

ai*#i  +  #2*^1  =  J>       #i*fl2  +  G-2*&2  =  °> 

^i*«i  +  &2*&i  =  o,     K*<h  +  b**b2  =  1, 
whence 


ox*     a2* 
b*    b* 


h    b2 


^1*^  +  a2*^2     ^1*^2  +  ^2*^2 
I         O 


ass  I. 


O       I 

1  We  call  a,  b  the  coefficients  of  the  number  a«i  +  bw2,  where  *t,  w2 
is  a  basis. 


THE   REALM    k(l). 


161 


Therefore 


=  ±  i. 


=  ±  i, 


The  condition  is  also  sufficient;  for,  solving  i)   for  tox  and  <o2, 
we  have,  if  2)  be  satisfied, 

co1  =  ±  (^2wi*  —  ^2W2*)> 

and  hence,  if  w,  =  c^  +  C2W2>  be  any  integer  of  the  realm, 

o>=±  (cj>a  +  c2bx)<»*  qz  (cxa2  +  c&)*f ; 

that  is,  w  =  dxiox*  +  ^2w2*j 

where  Jx  and  d2  are  rational  integers.  Since  there  is  an  infinite 
number  of  different  sets  of  rational  integers  ax,  a2,  bx,  b2  which 
satisfy  the  relation 

ax    a2 

K     b2 

there  is  an  infinite  number  of  bases  of  k(i). 

§4.     Discriminant  of  &(f). 

The  squared  determinant 


formed  from  any  basis  numbers  and  their  conjugates  is  called  the 
discriminant  of  the  realm,  and  is  denoted  by  d. 

That  d  is  the  same,  no  matter  what  basis  is  taken,  is  evident 
from  the  last  paragraph. 

For  if  wx,  <o2  and  w^*,  =  ax<ox  -j-  a2w2,  <o2*,  =  bxo)x-\-  b2o>2,  be  any 
two  bases,  then 

I  wi*     wi 

«>!*  ft). 


0 

a1<o1  + 

a2o)2 ,  bx<ax  -\-  b2<o2 

2 

ax<ox  -f-  a2(a2,  bx<Dx  -\-  b2<o2 

ax    a2 

2  1 

<ax       w2 

2 

wl 

o>2 

K     b2 

t          t 

I    0)x        w2 

«l' 

co2 

Hence,  since  1,  i  is  a  basis  of  k(i) 
d  = 


=  —  4- 


11 


1 62  THE   REALM    k(i). 

It 

that 


32  THE   REALM    k(l). 

It  is  easily  seen  that  if  w^wg  be  any  two  integers  of  k(i)  such 
tat 


2 

=  d, 


then  <*>!,  w2  is  a  basis  of  k(i). 

For  example : 

i  +  *    3  +  2*  I2 


—  4: 


i— i    3  — 2i  \ 

Hence  1  +  *,  3  +  2*  is  a  basis  of  &(0  as  we  have  already  seen. 
§5.    Divisibility  of  Integers  of  k(i). 

Any  integer,  a,  is  said  to  be  divisible  by  an  integer,  (3,  when 
there  exists  an  integer,  y,  such  that 

a  =  Py. 

We  say  that  /?  and  y  are  divisors  or  factors  of  a,  and  that  a  is 
a  multiple  of  /?  and  y. 

Ex.  i.    We  see  that  8  +  i  is  divisible  by  3  +  21,  since 

8  +  /=  (3  +  20(2-0. 

Ex.  2.     On  the  other  hand  5  +  2i  is  not  divisible  by  1  +  3*',  for  there 
exists  no  integer  of  k{i)  which  multiplied  by  1  +  3*  gives' 5 +  2*. 
This  can  be  shown  as  follows: 

If  we  set  s  +  2»=s  (l4-30(*Hhy0i  1) 

we  obtain  jr=r|^,   37  =  — |f  ; 

that  is,  there  are  no  integral  values  of  x  and  y  for  which  1)   will  hold. 

Hence  5  +  3*  is  not  divisible  by  1  +  31. 
This  can  also  be  shown  as  follows: 

5  +  2*       (5  +  2p(i— 30  =  n_  ii  v 
i+3*'       (i  +  30d-30       "       10 

As  immediate  consequences  of  the  above  definition  we  have  the 
following : 

i.  If  a  be  a  multiple  of  (3  and  ft  be  a  multiple  of  y,  a  is  a  mul- 
tiple of  y,  or  more  generally 

ii.  //  each  integer  of  a  series  a,(3,y,8,---,bea  multiple  of  the 
one  next  following,  each  integer  is  a  multiple  of  all  that  follow  it. 


THE   REALM    k(i).  1 63 

iii.  //  two  integers,  a  and  /?,  be  multiples  of  y,  then  a£  +  fa  is 
a  multiple  of  y,  where  $  and  t]  are  any  integers  of  the  realm. 

It  will  be  observed  that  iii  depends  not  only  upon  the  above 
definition  but  upon  the  fact  that  the  sum,  difference  and  product 
of  any  two  integers  of  k(i)  is  an  integer  of  k(i).  If  a  be  divis- 
ible by  /?,  then  a'  is  divisible  by  (3' ;  for,  if  a  =  £y,  then  a'  =  0'y'. 
In  particular,  if  a  rational  integer  be  divisible  by  any  integer  of 
k(i),  it  is  divisible  by  its  conjugate. 

Theorem  3.  If  a  be  divisible  by  /?, n[a]  is  divisible  by  n[(3]. 
For,  if  a  =  fiy,  it  follows  from  Th.  1  that 

n[a]=n[p]n[y], 

and  hence  that  n[a]  is  divisible  by  n[(3]. 

The  converse  of  this  theorem  is  not  in  general  true,  as  may  be 
seen  from  the  following  example: 

If  a  =  8-f-*'  and  £  =  3  —  2t,  n[a],  =  65,  is  divisible  by 
n[(3],  =  13,  but  a  is  not  divisible  by  /?;  for  putting 

8+t=  (3— *0(*+'j9» 

we  obtain  fractional  values  for  x  and  y. 

The  determination  of  the  conditions  under  which  n[a]  divisible 
by  n[fi]  is  a  sufficient  as  well  as  necessary  condition  for  a  to 
be  divisible  by  ft  must  be  postponed  until  the  unique  factoriza- 
tion theorem  has  been  proved  for  the  integers  of  k(i). 

If  two  or  more  integers,  a,  /?,  y,  •  •  • ,  of  k  (i)  be  each  divisible 
by  an  integer  fx  of  k(i),  fx  is  said  to  be  a  common  divisor  of 
a,P,y,  •••- 

§6.     Units  of  k(i).    Associated  Integers. 

We  have  seen  that  in  the  rational  realm  there  are  certain  in- 
tegers, zh  1,  called  units,  which  are  divisors  of  every  integer  of 
the  realm.  Evidently  ±  1  have  this  property  in  k(i),  and  are 
therefore  called  units  of  k(J).  We  ask  now  whether  there  are 
any  other  integers  of  k(i)  which  enjoy  this  property.  If  there 
be  such  integers  they  must  be  divisors  of  1,  and  conversely  every 
divisor  of  1  is  a  unit.     Let  e,  =  x  -\-yi,  be  a  unit  of  k(i)  ;  then 

ae=i,  1) 


164  THE    REALM    k(i). 

where  a  is  an  integer  of  k{i).    It  follows  that 

w[a]w[e]  =  1, 
and  hence  w[c]  =  1 ;  that  is, 

x*  +  y2=i.  2) 

That  n[e]  =1  is  not  only  a  necessary  but  also  a  sufficient  con- 
dition that  e  shall  be  a  unit,  is  evident  from  the  fact  that  from  it 

follows  ee  =  1 , 

and  hence  that  €  is  a  divisor  of  1. 
From  2)  .it  follows  that 

x=±  1,  ;y  =  o;  x  —  o,  y=:±i, 

and  hence  e  =  1,  —  1,1  or  —  i, 

Therefore  I, —  i,i,  —  i  are  the  units  of  k(i).  That  all  these  in- 
tegers are  units  of  k{i)  may  easily  be  verified,  since,  if  a  -j-  bi  be 
any  integer  of  k(i),  we  have 

a-|-  bi=i(a-\-bi) 

=  —  i( — a  —  bi) 

=*( — ai-\-  b) 

=  —  i(ai  —  b) 

Starting  with  the  original  definition  of  a  unit  as  an  integer 
which  is  a  divisor  of  every  integer  of  the  realm,  we  obtain  there- 
fore the  three  following  equivalent  definitions  for  the  units 
of  k(i): 

i.  They  are  the  divisors  of  1. 

ii.  They  are  those  integers  whose  reciprocals  are  integers. 
Hence  the  reciprocal  of  a  unit  is  a  unit. 

iii.  They  are  those  integers  whose  norms  are  1.  Hence  the 
conjugate  of  a  unit  is  a  unit. 

Two  integers,  a  and  (3,  with  no  common  divisor  other  than  the 
units  are  said  to  be  prime  to  each  other. 

It  is  customary  also  to  say  that  two  integers,  whose  common 
divisors  are  units,  have  no  common  divisor.     A  system  of  in- 


THE   REALM    k(t).  165 

tegers,  ax,  a2,  •••,a„,  such  that  no  two  of  them  have  a  common 
divisor  other  than  the  units  are  said  to  be  prime  each  to  each. 

As  in  the  rational  realm,  two  integers,  m  and  — m,  that  differ 
only  by  a  unit  factor,  are  said  to  be  associated,  so  in  k(i)  the 
four  integers,  a,  —  a,  ia  and  —  ia,  obtained  by  multiplying  any 
integer,  a,  by  the  four  units  in  turn,  are  called  associated  integers. 
For  example,  the  four  integers  3  +  21',  —  3  —  2i,  —  2  -f-  3*,  2  —  3J 
are  associated.  We  say  also  that  a,  — a,  ia,  — ia  are  the  asso- 
ciates of  a.  Any  integer  that  is  divisible  by  a  is  also  divisible  by 
—  a,  ia  and  — ia.  Hence  in  all  questions  of  divisibility  associated 
integers  are  considered  as  identical.  It  will  be  understood  from 
now  on  that  when  two  factors,  a,  (3,  of  an  integer  of  k(i)  are 
said  to  be  the  same,  they  are  merely  associated;  that  is,  a  =  e/?, 
where  c  is  a  suitable  unit.  They  may  or  may  not  be  equal,  equality 
.being  understood  in  the  ordinary  sense ;  that  is, 

a1  +  bxi  =  a2  +  b2i, 

when  and  only  when  ax  =  a,,  and  bt  =  b2. 

If  each  of  two  integers  be  divisible  by  the  other,  they  are  asso- 
ciated, for  let  a/ft  =  y,  then  p/a=i/y.  If  now  both  y  and  1/7 
be  integers,  then  y  is  a  unit  and  a  and  (3  are  associated. 

§  7.    Prime  Numbers  of  k(i). 

An  integer  of  k(i),  that  is  nut  a  unit  and  that  has  no  divisors 
other  than  its  associates  and  the  units,  is  called  a  prime  number 
ofk(i). 

An  integer  of  k(i)  with  divisors  other  than  its  associates  and 
the  units  is  called  a  composite  number. 

It  will  be  observed  that  these  definitions  are  identical  with  the 
corresponding  ones  in  the  rational  realm.  To  ascertain  whether 
any  integer  a,  not  a  unit,  is  a  composite  or  prime  number,  we  have 
only  to  determine  whether  or  not  a  can  be  resolved  into  two 
factors  neither  of  which  is  a  unit. 

We  put  therefore  a  =  (a -\- bi)  (c -\- di)  and  determine  for 
what  sets  of  integral  values  of  a,  b,  c  and  d  this  equation  is  sat- 
isfied. If  any  one  of  these  sets  of  values  be  such  that  neither 
a+  bi  nor  c  +  di  is  a  unit,  a  is  a  composite  number;  but,  if  for 
every  set  of  values  one  of  these  factors  be  a  unit,  a  is  a  prime. 


1 66  THE   REALM    k(i). 

Ex.  I.  To  determine  whether  3  is  a  prime  or  composite  number 
of  k(i). 

Put  3=(a  +  bi)  (c  +  di)  ; 

then  9  =  (c?  +  b2)(c2  +  d2), 

whence  we  have  either 

2        .  F1)    or  H 

C2+rf2  =  3J  C2+rf2  =  9J 

Remembering  that  a,  b,  c  and  d  must  be  rational  integers,  we  see  that  1) 
is  impossible,  while  from  2)  a-\-bi  is  a  unit.  Therefore  3  is  a  prime 
number  of  k(i). 

Ex.  2.  To  determine  whether  7  +  41"  is  a  prime  or  composite  number 
of  k(i). 

Put  7  +  4t'=(a  +  bi)  (c  +  dt)  ; 

then  65=  (a2  +  b2)(c2  +  d2), 

whence  we  have  either 

a2  +  b2=5    l  a2+fc2=i 

'2) 


Li)     or  I 

c2  +  <f=i3  j  <*  +  <**  =  65  j 

low  that  a  +  bi  is  a  unit,  but 

=  ±  I,  1  fl=±  I,    &=±2,  ] 

=  ±  2,  )  C=±2,    rf  =  ±  3,    J 


From  2)   it  would  follow  that  a  +  bi  is  a  unit,  but  1)   gives 

a=±  2,  b  =  ±  1,  )  a=±  1,  b 
c  =  ±  3,  d 

whence                      o  +  &*"  =  ±  (2  +  *)  or    ±  (1 — 2/),                             3) 

or                              a  +  bi  =  ±  (2  —  t)  or     ±  ( 1  -f-  21) ,                             4) 

and                            c  +  dt  =  ±  (3  +  2/)  or     ±(2  —  3?),                            5) 

or                               c  -j-  di  =  ±  (3  —  2O  or    ±  (2  +  30 >                           6) 

the  four  integers  after  each  sign  of  equality  being  associated. 

It  will  be  observed  that  this  process  gives  us  not  only  the  divisors 
of  7  +  4i  and  its  associates,  but  also  the  divisors  of  every  other  integer 
whose  norm  is  65;  that  is,  of  7  —  4*',  8  +  t,  8  —  *,  and  their  associates. 

Each  one  of  the  eight  values  of  a  +  bi  multiplied  by  any  one  of  the 
eight  values  of  c  -J-  di  gives  an  integer  whose  norm  is  65,  and  these  sixty- 
four  integers  fall  into  four  classes  of  sixteen  each  according  to  the  one 
of  the  integers  7  +  Ah  7  —  4h  8  +  i,  8  —  i,  with  which  they  are  as- 
sociated. Each  associate  of  each  one  of  these  four  integers  will  be 
repeated  exactly  four  times. 

Selecting  by  trial  the  divisors  of  7  -f-  4*,  we  see  that  any  integer  from 
4),  multiplied  by  a  suitable  one  from  6),  gives  7  +  4*. 

Thus  7  +  4*'=  (2  —  0(2  +  30-  7) 

Hence  7  +  4/  is  a  composite  number. 


THE   REALM    k(l).  1 67 

We  have  also,  7  +  41  =  (—  2  +   •)  (—  2  —  3*') , 

=  (      i  +  2*')(      3  —  2»), 

=  (—  I"  2*)(—  3  +  2*), 

but  these  factorizations  are  looked  upon  as  in  no  way  different  from 
7)  since  the  corresponding  factors  are  associated.  Hence  7  +  4*  can  be 
factored  in  only  one  way  into  two  factors,  neither  of  which  is  a  unit. 
If  now  we  attempt  to  factor  2  —  i  and  2  +  3*,  we  find  that  they  are 
prime  numbers,  and  hence  we  say  that  7  +  4*  has  been  resolved  into  its 
prime  factors. 

Ex.  3.     Resolution  of  —  23  +  41*'  into  prime  factors. 

If  we  endeavor  to  resolve  —  23  +  411  into  two  factors  neither  of 
which  is  a  unit,  we  find  that  it  can  be  done  in  seven  different  ways;  that  is, 

—  23  +  41*=  (1+3*)  (    10  +nt),  "" 
=  (i  +  5*)(      7+   60, 
=  (3  +  5*)(      4+    70, 
=  (l+  *')(     9  +  320,    \  8) 

=  (2+  *')(— 1  +  21O, 

=  (3  +  2*')(      1  +  13O, 

=  (4+  0(— 3  +  nO-  _ 

We  find,  however,  that  in  each  case  either  one  or  both  of  the  factors 
is  composite  and  we  resolve  the  composite  ones  into  the  following  factors 
all  of  which  can  easily  be  proved  to  be  prime: 

i-f  3»=  (i  +  0(2  +  0;  i+Sf=  (i  +  0(3+2*'); 

3  +  5*"=  (i  +  0(4  +  0  J  10+11*'  =  (3  +  2*')(4  +  0  ; 

7  +  6*=  (2  +  0(4  +  0;  4  +  7*  =  (2  +  0(3  +  2*'). 

when  these  values  are  substituted  in  8)  we  have  in  all  seven  cases 

—  23  +  41*  =  ( 1  +  0  (2  +  0  (3  +  »)  (4  +  0 ; 

that  is,  if  —  23  +  41 1  be  resolved  into  factors  all  of  which  are  prime, 
the  resolution  can  be  affected  in  only  one  way. 

It  is  now  evident  that  we  can,  as  in  the  case  of  the  rational 
integers,  represent  every  integer  of  k(i)  as  a  product  of  its  prime 
factors,  and  the  last  example  renders  it  probable  that  the  repre- 
sentation will  be  unique.  We  shall  proceed  to  prove  three 
theorems  which  will  enable  us  to  show  that  the  integers  of  k(i) 
have  indeed  this  all-important  property. 

§  8.     Unique  Factorization  Theorem  for  k(i). 

Theorem  A.  //  a  be  any  integer  of  k(i),  and  (3  any  integer  of 


1 68  THE   REALM    k(i). 

k(i)  different  from  o,  there  exists  an  integer  fi  of  k(i)  such  that 

n[a  —  fjL/3]<n[p]. 

Let  a/(3  =  a+bi, 

where  a  =  r-\-rlf  b  =  s  -\- s1}  r  and  ^  being  the  rational  integers 
nearest  to  a  and  b  respectively,  and  hence 

We  shall  show  that  /*,  =r-{-si,  will  fulfill  the  required  con- 
ditions. 

Since  a/p  —  /*  =  rt  +  *i*, 

whence  « [a//3  —  fi]  <  I ; 

or,  multiplying  by  «•[£], 

n[a  —  lip]  <n[p]. 

Ex.     If  a  3=  5  +  2i,  and  0  =  i  -f-  3/, 

then  a  —  5  +  2*'  _  1 1  _  1 3  i 

and  /*  =  1  — i, 

therefore  a  —  /*)3  =  5  -j-  2*  —  ( 1  —  i)  ( 1  +  3O  =  1, 

and  w[i]  <ra[i  -J- 3*]. 

The  method  given  above  for  selecting  /x  evidently  determines 
it  uniquely  unless  either  one  or  both  of  the  quantities  |^i|,  |^i|  be 
J,  in  which  cases  there  are  respectively  2  or  4  integers  which 
satisfy  equally  the  method  of  selection. 

There  are,  however,  values  of  /x  that  satisfy  the  requirements  of 
the  theorem  other  than  the  one  selected  as  above.  In  the  ex- 
ample given  above  it  would  serve  as  well  to  take 

H  =  2  —  i  or  1  —  21 ; 

for  5_L.2;_(2_;)(I_|_3f)==_3*; 

and  n[  —  p]  <fl[l+3*]  \ 

likewise  5  +  2* —  (.1  —  2*)  ( 1  +  3*)  = — 2  +  i, 

and  n  [ —  2  +  *']  <  n  [  1  +  3*] . 


THE   REALM    k(i). 


169 


It  can  be  easily  shown  that  there  are  in  general  (including  the 
one  selected  as  in  the  proof)  two,  three  or  four  values  of  fi  which 
satisfy  the  requirements  of  the  theorem.  The  particular  value  of 
fi  selected  as  above  may  be  called  the  nearest  integer  to  a/ft. 

The  other  possible  values  of  n  are  found  among  the  integers 
rt  +  sai  such  that  r2,s2  differ  respectively  from  rl,sl  by  1. 

This  will  be  made  clearer  by  a  graphical  proof  of  the  theorem 
to  which  we  are  led  by  its  statement  in  the  following  form : 

//  a/p  be  any  number  of  k(i),  there  exists  an  integer  /x  of 

k(i)  such  that  n[a//3  —  fi]  <  I. 


Y 

-2+2i 

-l+2i 

2i 

l+2i 

2+2i 

-2+i 

-1  +  i 

i 

1  +  i 

2+i 

■■»■ 

-2 

-1 

0 

1 

2 

/^ 

P,                                        Po 

-2-i 

-1-i 

-i 

1                      1-i 

2-i 

/ 

-2-2i 

-l-2£ 

-2i 

\l-2i 

p          y 

2-2i 

Representing  as  is  usual  the  number  x -\- yi  by  a  point  whose 
coordinates  referred  to  rectangular  axes  are  x  and  y,  we  see  that 
the  integers  of  k(i)   are  the  points  of  intersection  of  a  lattice 


170  THE   REALM    k{%). 

formed  by  two  systems  of  straight  lines  parallel  respectively  to  the 
axes  of  x  and  y,  and  at  the  distance  1  apart.1 

Our  problem  is,  given  any  number  y  of  k(i),  we  are  required  to 
find  all  integers,  ft,  of  k(i)  such  that 

»[y— rf<i-  1) 

Let  G  and  N  be  points  representing  the  numbers  y,  =  a  -\-  hi, 
and  v,  =c  -f-  df,  respectively;  then  every  number,  v,  of  k(t)  such 

that  ^.[y  —  v]  <  1 

is  represented  by  a  point  lying  within  the  circle  of  radius  1  de- 
scribed about  G  as  a  center,  and  conversely  every  number,  v,  of  the 
realm  represented  by  a  point  lying  within  this  circle  satisfies  1)  ; 

for  (x  —  ay+  (y  —  b)2=i 

is  the  equation  of  a  circle  of  radius  1  with  center  at  G,  and  we  have 

(c-ay+(d-by<i; 

that  is  n[y  —  v]  <  1 

when  and  only  when  the  point  (c,d)  lies  within  this  circle. 

The  graphical  solution  of  our  prpblem  consists  therefore  merely 
in  describing  a  circle  of  radius  1  around  the  point  representing  y 
and  observing  what  lattice  points  fall  within  it. 

In  the  figure  the  point  G  represents  the  number  y  =  |£  —  ^§* 
(see  example  above),  and  a  circle  of  radius  1  described  around 
G  as  a  center  is  seen  to  enclose  the  three  points  Px,  P2,  P3,  repre- 
senting the  integers  1  —  i,  2  —  i,  1  —  2L  Moreover,  no  other  in- 
teger point  falls  within  this  circle. 

The  integers  1  —  i,  2  —  i,  1  —  2»  are  all  the  values  of  fi  which 

satisfy  the  condition  n[y  —  /a]  <  1, 

the  integer  1  — i,  which  is  the  one  given  by  the  method  of  selec- 
tion used  in  the  proof,  being  represented  by  the  lattice  point  near- 
est to  G. 

It  is  evident  that  the  only  possible  values  of  fi  are  those  repre- 
sented by  the  vertices  of  the  lattice  square  in  which  the  point  G, 
representing  y,  lies. 

'Cahen:  p.  357. 


THE   REALM    k(i).  I7I 

We  see  that  two,  three  or  four  of  these  vertices  will  satisfy  the 
required  condition  according  as  G  lies  in  the  unshaded,  lightly 
shaded  or  heavily  shaded  portions  of  the  square,  the  square  being 
thus  partitioned  by  describing  from  each  vertex  as  a  center  an 
arc  of  a  circle  of  radius  i. 

Glt  G  and  G2  illustrate  respectively  the  first,  second  and  third 
cases.  G±  and  G2  illustrate  also  the  cases  in  which  there  are  re- 
spectively two  or  four  equally  near  lattice  points  (original  method 
of  selection  is  not  unique). 

Returning  once  more  to  the  theorem  in  its  original  form,  we 
observe  that  it  is  equivalent  to  saying  that  for  every  integer  /?, 
different  from  o,  considered  as  a  modulus  there  exists  a  complete 
residue  system  such  that  the  norms  of  all  the  integers  composing 
this  system  are  less  than  n[/3]. 

This  interpreted  graphically  implies  that  if  we  describe  around 
the  origin  a  circle  with  radius  equal  to  "\/n>[p],  that  is,  passing 
through  the  point  representing  /?,  there  will  be  among  the  integers 
represented  by  the  lattice  points  lying  inside  this  circle  a  complete 
residue  system,  modulus  p. 

Theorem  A  is  equivalent  to  saying  that  we  can  divide  a  by  (3 
so  as  to  obtain  a  remainder  whose  norm  is  less  than  n[(3],  the 
quotient  being  fi.  In  this  form  its  analogy  with  Theorem  A  in  R 
is  even  more  clearly  brought  out.  It  enables  us  to  do  for  k{%) 
exactly  what  we  did  in  R  by  means  of  Theorem  A ;  that  is,  by  an 
algorithm  strictly  analogous  to  that  used  in  R  to  find  a  common 
divisor,  8,  of  any  two  integers  a  and  /?,  such  that  every  common 
divisor  of  a  and  (3  divides  8.  In  other  words,  it  enables  us  to 
prove  that  any  two  integers  of  k(i)  have  a  greatest  common 
divisor  and  to  find  it.1 

For  example;  let  the  two  integers  be  112  -f- *  and  — 57  +  79?'. 

We  have      II2  +  *  .  =  "^5 -8905/     whence  „  =  _  ,  _  ,- 
—  57  +  79*  9490 

and  1 12  +  i  —  (—  1  —  i)  (—  57  -f-  79J)  =  —  24  +  23?'. 

xSee  Dirichlet-Dedekind :  p.  439. 


172  THE   REALM    k{l). 

Likewise  57  ~r  9l  __  31  5     5  5*>  whence  ih.  =  3  —  h 

—  24  +  231  1 105 

and        —57+  79*  —  (3  — 0  (—  24  -\-23i)  =  —  8  —  14*. 

Likewise  -  *4  +  23*  =  -  130  -  5*»  f  whence  fJ^  =  _1_2if 
—  8  —  14*      260 

and       —  24  -f  23*  —  (  —  1  —  2»)  ( —  8  —  14*)  as  —  4  —  71. 

Finally  ~~    ~  I4f  —  2,  whence  ft  =  *M      f*, 

—  4  —  71 

and  — 8  —  V4» — (2)  (  —  4  —  7*)  =  o. 

Therefore  — 4  —  71  is  the  greatest  common  divisor  of  112  +  /  and 
—  57  +  79*- 

Instead,  however,  of  proving  the  existence  of  a  greatest  common 
divisor  of  any  two  integers  of  k(i),  we  shall  proceed  as  in  R, 
and  shall  prove  the  following  theorem  of  which  the  greatest  com- 
mon divisor  theorem  is  an  immediate  consequence. 

Theorem  B.  If  a  and  p  be  any  two  integers  of  k(i)  prime  to 
each  other,  there  exist  two  integers,  £  and  77,  of  k(i)  such  that 

a£  +  pv=i. 

If  either  a  or  /?  be  a  unit,  the  existence  of  the  required  integers, 
i,  7),  is  evident.  We  shall  now  show  that,  if  neither  a  nor  p  be  a 
unit,  the  determination  of  £  and  rj  can  be  made  to  depend  upon 
the  determination  of  a  corresponding  pair  of  integers  £lf  i/x,  for 
a  pair  of  integers,  alf  plf  prime  to  each  other  and  such  that  the 
norm  of  one  of  them  is  less  than  both  n[a]  and  n[p]. 

Assume  n[p]  ^n[a],  which  evidently  does  not  limit  the  gen- 
erality of  the  proof. 

By  Th.  A  there  exists  an  integer  /*  such  that 

n[a  —  iip]  <n[p]. 

Then  p  and  a  —  /*/?  are  a  pair  of  integers,  alt  pi3  prime  to  each 
other  and  n[a  —  fxp]  is  less  than  both  n[a]  and  n[p~\. 
If,  now,  two  integers,  £lf  rf19  exist  such  that 

that. is,  £&  +  (a  —  iiPhx=i, 


THE   REALM    k(i).  1 73 

we  have  0%  +  p(i1  —  ^J  =  i, 

and  hence  £  =  ,>?i>     77==£i —  Mi- 

The  determination  of  &,  ^  for  at,  ^  may,  if  neither  ax  nor  /^ 
be  a  unit,  be  made  to  depend  similarly  upon  that  df  £2>  V2  f°r  a 
pair  of  integers  a2,  fi2  prime  to  each  other  and  such  that  the  norm 
of  one  of  them  is  less  than  both  n[a±]  and  w-[/y. 

By  a  continuation  of  this  process,  we  are  able  always  to  make 
the  determination  of  |  and  rj  depend  eventually  upon  that  of  £n,  rjn 
for  a  pair  of  integers  an,  /?«,  one  of  which  is  a  unit. 

Since  the  existence  of  |M  and  rjn  is  evident,  the  existence  of  | 
and  r]  is  proved. 

,  We  shall  see  later  that,  although  the  proof  here  given  of  the 
unique  factorization  theorem  depends  upon  Th.  A,  there  are 
realms  in  which  the  unique  factorization  theorem  holds  but  Th. 
A  does  not  hold.  However,  we  shall  see  also  that  each  of  the 
three  theorems  B,  C  and  the  unique  factorization  theorem  is 
necessary  and  sufficient  for  the  validity  of  the  other  two. 

Cor.  1.  If  a  and  ft  be  any  two  integers  of  k(i),  there  exists  a 
common  divisor,  8,  of  a  and  ft  such  that  every  common  divisor  of 
a  and  ft  divides  8,  a>nd  there  exist  two  integers,  $  and  rj,  of  k(i) 

such  that  ai-{-  firj  =  8. 

The  proof  is  the  same  as  in  R. 

We  call  8  the  greatest  common  divisor  of  a  and  /?. 

Cor.  2.  //  alt  a2,  •••,a„  be  any  n  integers  of  k(i),  there  exists 
a  common  divisor,  8,  of  alt  a2,  •  •  • ,  an  such  that  every  common 
divisor  of  at,  a2,  •  •  • ,  an  divides  B,  and  there  exist  n  integers 
£i»£»>  ••*,lii    such  that 

a£t  +  a2i2  H \-  an$n  =  8. 

Theorem  C.  //  the  product  of  two  integers,  a  and  (3,  of  k{i) 
be  divisible  by  a  prime  number,  *,  at  least  one  of  the  integers  is 
divisible  by  ir. 

Let  a/3  =  y7T,  where  y  is  an  integer  of  k(i),  and  assume  a  not 


174  THE   REALM    k(i). 

to  be  divisible  by  ar.  Then  a  and  tt  are  prime  to  each  other  and 
there  exist  two  integers,  £  and  rj,  of  k(i)  such  that 

a£  +  7n?=i.  2) 

Multiplying  2)  by  p,  we  have 

and  therefore  7r(y|  +  /fy)  =  /3, 

where  y|  +  ft  1S  an  integer  of  &(*)  ;  hence  (3  is  divisible  by  tt. 

Cor.  1.  //  the  product  of  any  number  of  integers  of  k(i)  be 
divisible  by  a  prime  number,  ir,  at  least  one  of  the  integers  is  divis- 
ible by  ir. 

Cor.  2.  If  neither  of  tzuo  integers  be  divisible  by  a  prime  num- 
ber, tt,  their  product  is  not  divisible  by  v. 

Cor.  3.  //  the  product  of  two  integers,  a  and  /?,  be  divisible 
by  an  integer,  y,  and  neither  a  nor  (3  be  divisible  by  y,  then  y  is  a 
composite  number. 

Theorem  4.  Every  integer  of  k(i)  can  be  represented  in  one 
and  only  one  way  as  the  product  of  prime  numbers. 

Let  a  be  an  integer  of  k(i).     If  a  be  not  itself  a  prime  number, 

we  have  a  =  py,  3) 

where  ft  and  y  are  integers  of  k(i)  neither  of  which  is  a  unit. 

From  3)  it  follows  that  n[a]  =  n[fi]n[y] ,  whence,  since 
n[p]  =j=  1  and  n[y]  =)=i,  we  have  n[fi]  and  n[y]  <  n[a]. 

liftbe  not  a  prime  number,  we  have  as  before 

P— An, 

where  px  and  yx  are  integers  neither  of  which  is  a  unit,  and  hence 
7t[/?J  and  n[y±]  <  n[p].  If  px  be  not  a  prime  number,  we  pro- 
ceed in  the  same  manner,  and,  since  n[p],  *[&],  n[P2],  '"  form 
a  decreasing  series  of  positive  rational  integers,  we  must  after  a 
finite  number  of  such  factorizations  reach  in  the  series  p,  pit  p2,  •  •  • 
a  prime  number  wv     Thus  a  has  the  prime  factor  ttx,  and  we  have 

a  =  7rxax. 


THE   REALM    k(t).  1 75 

Proceeding  similarly  with  ax,  in  case  it  be  not  a  prime  number, 
we  obtain  a1  =  7r2a2, 

where  tt2  is  a  prime  number,  and  hence 

OL  =  ttxtt2OL2. 

Continuing  this  process  we  must  reach  in  the  series  a,  alf  a2,  •  •  • 
a  prime  number  tt„,  since  n[a],  n[ax],  n[a2],  ••  •  form  a  decreas- 
ing series  of  positive  rational  integers.     We  have  thus 

CL TTxTTc,TT%   '  '  '  7Tfi) 

where  the  tt's  are  all  prime  numbers ;  that  is,  a  can  be  represented 
as  a  product  of  a  finite  number  of  factors  all  of  which  are  prime 
numbers. 

It  remains  to  be  proved  that   this   representation   is   unique. 

Suppose  that  a  =  pxp2p3  •  •  •  pm 

is  a  second  representation  of  a  as  a  product  of  prime  factors.  It 
follows  by  Th.  C,  Cor.  i  from 

iri",2ir3  * '  *  **■  ~  P1P2PZ  ' ' '  pm,  4) 

that  at  least  one  of  the  p's,  say  plf  is  divisible  by  vu  and  hence 
associated  with  7r±;  that  is,  p1  =  e17r1,  where  cx  is  a  unit.    Dividing 

4)  by  ttx,  we  have       tt2tt3  •  •  •  irn  =  exp2p3  ■  ■  •  pm. 

From  this  it  follows  that  at  least  one  of  the  remaining  p's,  say  p2, 
is  divisible  by  ir2,  and  hence  associated  with  it.  Thus  p2  =  e2Tr2, 
where  c2  is  a  unit,  and  hence 

7T3   •  •  •   7Tn  ==  Z\£oP3   '  '  '  pm- 

Proceeding  in  this  manner,  we  see  that  with  each  n  there  is 
associated  at  least  one  p,  and,  if  two  or  more  tt's  be  associated  with 
one  another,  at  least  as  many  p's  are  associated  with  these  tt's, 
and  hence  with  one  another. 

In  exactly  the  same  manner  we  can  prove  that  with  each  p  there 
is  associated  at  least  one  it,  and,  if  two  or  more  /o's  be  associated 
with  one  another,  at  least  as  many  tt's  are  associated  with  these 
p's,  and  hence  with  one  another. 


I76  THE   REALM    k(i). 

Hence  considering,  as  we  always  shall,  two  associated  factors 
as  the  same,  the  two  representations  are  identical ;  that  is,  if  in 
the  one  representation  there  occur  e  factors  associated  with  a 
certain  prime,  there  will  be  in  the  other  representation  exactly  e 
factors  associated  with  the  same  prime. 

We  can  now  evidently  write  every  integer,  a,  of  k(i)  in  the  form 

a  =  «-!**/■  •  •  ■  7Tnen, 

where  vlfv2,  »••,*«  are  the  unassociated  prime  factors  of  a,  and  c 
a  suitable  unit.     Moreover,  this  representation  is  unique. 

Cor.  1.  If  a  and  (3  be  prime  to  each  other  and  y  be  divisible 
by  both  a  and  /?,  then  y  is  divisible  by  their  product. 

Cor.  2.     //  a  and  (3  be  each  prime  to  y,  then  af3  is  prime  to  y. 

Cor.  3.  //  a  be  prime  to  y  and  a/?  be  divisible  by  y,  13  is  divis- 
ible by  y. 

We  have  seen  that  the  divisibility  of  n[a]  by  n[fi]  is  a  neces- 
sary condition  for  the  divisibility  of  a  by  /?.  We  shall  now  show 
that  it  is  only  when  either  a  or  J3  is  a  rational  integer  that  the 
condition  is  also  sufficient. 

Let  a  m  fc***^  -  */*,  0  =  %??(>?  -  f? 

be  representations  of  a  and  (3  as  products  of  powers  of  their  dif- 
ferent prime  factors,  rja  and  rj^  being  units. 

From  n[a]=m  •  n[/3], 

where  m  is  a  positive  rational  integer,  it  follows  that 

*i  JL2        'Lk      '4     ''2  "k  In    "1   "2        ri     r\    rz         ri    > 

from  which  we  see  that  each  prime,  pi}  of  the  set  p1}p2,  '•',pl  is 
associated  with  one  of  the  7r's  or  with  one  of  the  tt"s,  say  vj  or  «■/, 
and  that  ri  3>  pj.  In  order  that  a  may  be  divisible  by  /?  we  must 
have  every  p  associated  with  an  unaccented  ?r,  which  will  not  be 
in  general  the  case.  When,  however,  a  is  a  rational  integer  we 
have  a  =  a',  and  this  condition  is  satisfied,  and  hence  /?  divides  a. 

If  0  be  a  rational  integer  it  is  easy  to  see  likewise  that,  when 
n[a]  is  divisible  by  n[(3],  a  is  divisible  by  (3. 


THE  REALM    k(i).  I  77 

§9.    Classification  of  the  Prime  Numbers  of  fe(i). 

Every  prime,  ir,  of  k(i)  divides  an  infinite  number  of  positive 
rational  integers;  for  example,  u[tt]  and  its  multiples.  Among 
these  positive  rational  integers  there  will  be  a  smallest  one,  p, 
and  p  will  be  a  rational  prime  number,  for  if  p  be  not  a  prime, 
that  is,  if  p=p1p2,  it  would  divide  either  px  or  p2,  and  hence  p 
would  not  be  the  smallest  rational  integer  that  -n  divides.  In 
order,  therefore,  to  find  all  primes  of  k(i)  we  need  only  examine 
the  divisors  of  all  rational  prime  numbers  considered  as  integers 
of  k(i). 

Moreover  it  is  evident  that  no  prime  of  k(i)  can  divide  two 
different  rational  primes,  for  then  it  would  divide  their  rational 
greatest  common  divisor,  I,  and  hence  be  a  unit.  Therefore  every 
prime  of  k(i)  occurs  once  and  but  once  among  the  divisors  of 
the  rational  primes  considered  as  integers  of  k(i). 

We  have  seen  already  that  there  are  rational  primes,  as  3, 
which  are  also  primes  of  k(i),  and  other  rational  primes,  as  5, 
which  are  factorable  in  k(i).  Denoting  then  by  p  the  smallest 
rational  prime  that  it  divides,  we  have 

p  =  ira,  1) 

and  hence  p2  =  n[ir]n[a]. 

We  have  then  two  cases 

.     f*H=/>,  ..    f *[«]**#*, 

\n[a]=p.  '    \n[a]  =  i. 

i.  From  n[ir]  =inr'==p  and  1)  it  follows  that  a  =  7r'.  If 
tt  =  a  +  bi,  we  have  then 

p  =  a2  +  b2. 

Assume  p  =f=  2 ;  then  either  a  or  b  must  be  odd  and  the  other 

even  and  therefore  />  =  i,  mod  4. 

Hence  when  a  positive  rational  prime  other  than  2  is  the  product 
of  two  conjugate  primes  of  k(i),  it  has  the  form  4n  +  i- 
When  p==2,  we  have 

2=(I+*)(I—  f), 

12 


I78  THE   REALM    k(J). 

and  hence  2=*{i — i)2 ; 

that  is,  2  is  associated  with,  and  hence  divisible  by,  the  square  of 
a  prime  of  k  (i) . 

ii.  Since  n[a]  =»I,  a  is  a  unit  and  hence  p  is  associated  with 
the  prime  ir;  that  is,  p  is  a  prime  in  k(i).  Hence  a  rational  prime 
p  is  either  a  prime  of  k(i)  or  the  product  of  two  conjugate 
primes  of  k(i). 

When  p  is  a  prime  of  the  form  4W —  1  it  is  always  a  prime  in 
k(i),  for  we  have  seen  that  p  is  factorable  into  two  conjugate 
primes  of  k(i)  only  when  it  is  2  or  of  the  form  4n  +  1. 

To  prove  now  that  every  rational  prime  of  the  form  4W  +  1  can 
be  represented  as  the  product  of  two  conjugate  primes  of  k(i) 
we  observe  that  from 

£s==I,  mod  4, 

it  follows  that  the  congruence 

.ar2  =  —  i,mod£, 
has  roots.     Let  a  be  a  root.     Then 

a2  =  —  1,  modp, 

and  hence  (a  -\-  i)  (a  —  »)  aso,  mod  p. 

Since  a-\-i  and  a  —  i  are  integers  of  k(i),  the  integer  p,  if  a 
prime  of  k(i),  must  divide  either  a  +  t  or  a  —  i.  This  is  how- 
ever impossible,  for  from 

a  ±  i=p(c  +  di), 

where  c  +  cfo'  is  an  integer  of  k(i),  it  would  follow  that  pd=±  1, 
which  can  not  hold  since  p  and  d  are  both  rational  integers  and 
p  >  I.  Hence  />  is  not  a  prime  in  &(*)>  and  since  the  only  way  in 
which  a  rational  prime  can  be  factored  in  k(i)  is  into  two  conju- 
gate prime  factors,  p  is  factorable  in  this  manner. 

Collecting  the  above  results,  we  see  that  the  primes  of  k(i) 
may  be  classified  in  the  following  manner,  according  to  the  rational 
primes  of  which  they  are  factors. 

1 )  All  positive  rational  primes  of  the  form  4%-\- 1  are  factor- 
able in  k(i)  into  two  conjugate  primes,  called  primes  of  the  first 
degree. 


THE  REALM    k(i).  1 79 

2)  All  positive  rational  primes  of  the  form  411 — 1  are  primes 
in  k(i),  called  primes  of  the  second  degree. 

3)  The  number  2  is  associated  with  the  square  of  a  prime  of 
the  first  degree. 

It  will  be  observed  that  the  norm  of  every  prime  tt  of  k(i)  is 
a  power  (first  or  second)  of  a  rational  prime  and  that  the  degree 
of  tt  is  the  exponent  of  this  power. 

Moreover,  we  notice  that  2  is  the  only  rational  prime  that  is 
divisible  by  the  square  of  a  prime  of  k(i)  ;  for,  if  this  were  true 
of  any  other  rational  prime  of  the  form  4»  +  1,  we  should  have 
tt  associated  with  tt',  and  hence 

a  -j-  bi  =  a  —  bi,  —  a  +  bi,  b  +  ai  or  —  b  —  ai, 

which  give  a  =  o,  b=^o,  or  a  =  =%=b,  all  of  which  are  seen  to  be 
incompatible  with  p  =  a2  +  b2. 

§  10.  Factorization  of  a  Rational  Prime  in  k(i)  determined 
by  the  value  of  (d/p). 

The  rational  primes  may  be  classified  with  regard  to  their 
factorization  in  k(i)  in  the  following  manner: 

1)  Those  of  which  the  discriminant  is  a  quadratic  residue  are 
factorable  into  two  conjugate  primes  in  k(i),  called  primes  of 
the  first  degree.  For  (d/p)  =  i  implies  p  =  ^n-\- 1,  since 
d  =  —  4,  and  we  have  seen  that  all  rational  primes  of  this  form 
are  thus  factorable  in  k(i). 

2)  Those  of  which  the  discriminant  is  a  quadratic  non-residue 
remain  primes  in  k(i),  called  primes  of  the  second  degree.  For 
(d/p)= — 1  implies  p  =  ^n  +  3,  and  we  have  seen  that  all 
rational  primes  of  this  form  remain  primes  in  k(i). 

3)  Those  which  divide  the  discriminant  {expressed  symbol- 
ically by  (d/p)  =0)  are  associated  with  the  squares  of  primes 
of  the  first  degree  in  k(i). 

Evidently  2  is  the  only  rational  prime  which  divides  the  dis- 
criminant of  k(i)  and  we  have  seen  that  2  =  i(i — i)2.  The 
following  table  expresses  the  above  results : 


©=■•>= 


l80  THE   REALM    k(i). 


3) 


(jh°'p= 


Ex.  Show  that,  if  a,  =a-\-bi,  be  any  integer  of  k(i),  such  that  a 
and  b  have  no  common  rational  divisor,  and  c  be  any  rational  integer 
divisible  by  a,  then  c  is  divisible  by  n[a]. 

§11.    Congruences  in  k(i). 

Exactly  as  in  the  case  of  rational  integers,  we  say  that  two 
integers  a,  (3,  of  k(i)  are  congruent  with  respect  to  the  modulus, 
/x,  if  their  difference  be  divisible  by  fi,  and  write 

a  =  p3  mod /a. 

The  laws  of  combination  that  were  proved  for  congruences  in 
R  hold  here. 

We  can  now  divide  all  integers  of  k(i)  into  classes  with  respect 
to  a  given  modulus,  li,  putting  two  integers  in  the  same  class  or 
different  classes,  according  as  they  are  or  are  not  congruent  to 
each  other,  mod  /x.  We  shall  show  that  for  any  given  modulus  n 
there  will  be  n[fi]  such  classes.  To  do  this  we  shall  need  the 
following  theorem : 

Theorem  5.  There  exist  among  the  multiples  of  any  integer 
fi,  of  k(i)  two,  t1,  =  oo»1,  i2,  =  bw1  +  C(o2,  such  that  every  multiple 
of  p  can  be  expressed  in  the  form 

where  a,  b,  c,  llt  l2  are  rational  integers  and  <d1}  w2  is  a  basis  of  k(i). 
Suppose  all  multiples  of  li  to  be  written  in  the  form 

1  =  a1b)1  -j-  a2<n2, 

and  consider  those  in  which  a2=|=o. 

Among  them  must  be  some  in  which  a2  is  smaller  in  absolute 
value  than  in  any  of  those  remaining. 

Let  t2,  =bo>1  -\~co)2,  be  one  of  these-;  then  c  Will  divide  the 
coefficient  a2  in  every  multiple  of  fi ;  for,  if  this  be  not  the  case, 

xThis  indicates  that  p  is  unfavorable  in  the  realm  under  discussion. 


THE   REALM    k(i).  l8l 

let  /?,  =  b1<o1  +  cxw2,  be  a  multiple  of  /x  such  that  cx  is  not  divisible 
by  c,  and  let  d  be  the  greatest  common  divisor  of  c  and  c4.  There 
exist  two  rational  integers  e,  elf  such  that 

ec  +  excx  =  rf, 
and  hence         y  =  £i2  +  eiP  =  (  *&  +  eJ>i )  <"i  +  ^2 
is  a  multiple  of  /x  in  which  a2  is  less  in  absolute  value  than  c,  but 
not  o.     But  this  is  contrary  to  our  original  hypothesis.     Hence 

we  have  a2  =  l2c, 

where  L  is  a  rational  integer,  and  hence 

t  —  l2i2  s=  (<*,  —  /2&  )  Wj. 

Consider  now  those  multiples  of  /*  in  which  a2=:o,  but  ax=%=o. 

There  will  be  some  among  them  in  which  ax  is  less  in  absolute 
value  than  in  any  of  those  remaining. 

Let  tp  =a<olf  be  one  of  these. 

It  is  seen  as  above  that  a  is  a  divisor  of  the  coefficient  ax  in 
every  multiple  of  p.  in  which  a2  =  o,  ^=4=0.  We  have,  therefore, 
since  (a^  —  l2b)<ox  is  a  multiple  of  fi  belonging  to  this  class, 

t  —  /2t2  =  (  ax  —  l2b  )  <ox  =  lxix, 

where  lx  is  a  rational  integer,  and  hence 

l  =  11l1jt12i2. 

Any  pair,  fix,ix2f  of  multiples  of  p,  such  that  every  multiple  of  fi 
can  be  written  in  the  form 

mxfix  +  m2ti2, 

where  mx,m2  are  rational  integers,  we  call  a  basis  of  the  mul- 
tiples Of  fl. 

The  pair  of  multiples  of  p,  a<ox,  b<ox  +  c&>2,  selected  as  above, 
and  in  which  in  addition  a  and  c  are  positive,  is  called  a  canonical 
basis  of  the  multiples  of  p. 

Theorem  6.  //  px,  fi2  be  a  basis  of  the  multiples  of  p,  the 
necessary  and  sufficient  condition  that 

H*  =  axfix  +  a2fi2, 
M2*  =  £i/*i  +  &2/*2> 


1*52 


THE   REALM    k  (t)  . 


zuhere  ax,  a2,  blt  b2  are  rational  integers,  shall  be  also  a  basis  of 
the  multiples  of  /jl  is 


a1     a2 
b<     bn 


=  ±  i 


The  proof  of  the  theorem  is  the  same  as  that  of  Th.  2. 
Theorem  7.     // 

fix  =  a1(a1  -J-  a2(ti2, 

fi2  =  b1o>1  +  b2<o2, 

be  any  basis  of  the  multiples  of  ft,  then 

bt   b2     "*W; 

It  is  evident  from  the  last  theorem  (see  proof  of  Th.  2)  that 
the  absolute  value  of  the  determinant 

ax     a0 

is  the  same  for  every  set  of  basis  numbers  of  the  multiples  of  p. 
Hence  we  need  only  determine  its  value  for  some  particular  basis. 

The  integers  fi  =  ax  -f-  a2h 

fxi  =  —  a2  -f-  a±i, 

constitute  a  basis  of  the  multiples  of  p,  and 

Hence  the  theorem  is  proved. 

Theorem  8.  //  fi  be  any  integer  of  k(i),  the  number  of  num- 
bers in  a  complete  residue  system,  mod  fi,  is  n[fi]. 

Let  cualt  b(dx  +  c<»2  be  a  canonical  basis  of  the  multiples  of  fi 
and  consider  the  system  of  integers 


(  u  =  o,  1,  "-,a — 1, 

U<at  -\-  Vu)9  < 

11         2    {  v=pO,  I,  ••-,£ — I, 


I) 


which  are  evidently  ac,  =n[fx],  in  number. 

We  shall  show  that  the  integers  1 )  constitute  a  complete  residue 
system,  mod  fi. 


THE   REALM    k(i).  1 83 

First,  each  of  them  is  incongruent  to  all  the  others,  mod  p.,  for 
if  fijttj  -\-  vxw2,  n2v>x  -\-  v2<o2  be  any  two  of  them,  and 

u1(a1  -\-  v-^2  sa  u2(o1  -f-  v2(o2,  mod  fi, 

then  (u1  —  u2)  <*>!  +  (yx  —  z/2)<o2=o,  mod /a, 

and  hence,  since  c  is  the  greatest  common  divisor  of  the  coeffi- 
cients of  w2  in  all  multiples  of  fi, 

vx  —  z/2==o,  mode. 

But  vt  and  v2  are  both  less  than  c;  hence 

vx  =  v2. 

It  follows  that  u1  —  w2  =  o,  mod  fi, 

and  hence,  since  a  is  the  greatest  common  divisor  of  the  coefficient 
of  (i)t  in  all  multiples  of  fx  in  which  the  coefficient  of  w2  is  o, 

ui  —  u2  —  °>  m°d a- 
But  ux  and  w2  are  both  less  than  a ;  hence 

ux  =  u2. 
Thus  w^i  -(-  v1o)2  =  u2o)1  -\-  v2<o2, 

and  the  numbers  i)  are  seen  to  be  incongruent  each  to  each, 
mod  fi.  Moreover,  every  integer  of  the  realm  is  congruent  to  one 
of  the  integers  i),  mod  ll.     For,  let 

0)  =  f  1<ttl  +  t2<o2 

be  any  integer  of  k(i),  and  let 

t2  =  mc  +  r2, 
where  m  and  r2  are  rational  integers  and  r2  satisfies  the  condition 

o  g  r2  <  c. 
Also  let  ft  —  mb=na-\-rx, 

where  n  and  r,  are  rational  integers  and  rx  satisfies  the  condition 

ogr1<ia. 
Then        txvx  -f-  t2o)2=  (mb  -f-  na  +  ri)wi  +  (mc  +  r2)<a2 

=  no*}  +  m{bux  +  c<o2)  -f-  rlWl  +  r2w2 ; 


184  THE   REALM    k(i). 

and  hence  t1o)1  +  1 2co2  m  r^!  -j-  f2<o2,  m°d  /a, 

where  r^  +  r2a>2  is  one  of  the  integers  1 ) .  Hence  every  integer 
of  the  realm  is  congruent,  mod  /a,  to  one  and  but  one  of  the 
integers  1). 

The  integers  1)  constitute,  therefore,  a  complete  residue  system, 
mod  fi,  and  being  n[p]  in  number  the  theorem  is  proved. 

We  can  construct  a  complete  residue  system  for  any  modulus, 
fi,  by  means  of  the  method  employed  in  the  above  proof.     Taking 

1,  i  as  a  basis,  we  let  n  =  m(p  +  qi), 

where  m  is  the  largest  rational  integer  that  divides  p,  p  and  q 
being  consequently  prime  to  each  other. 

It  is  easily  seen  that  m(p2-\-q2)  is  the  rational  integer  of 
smallest  absolute  value  divisible  by  fi;  that  is, 

a  =  m(p2  +  q2). 

Since  ac=n[fi]  —  m2(p2  +  q2), 

we  have  therefore      c  =  m.     q 
Hence  the  n[fi]  integers 

tt  =  o,i,  ••■,m(p2  +  q2)  —  1, 
v  =  o,  1,  '-'}m —  1, 

is  a  complete  residue  system,  mod  /*. 

Ex.    Let  fi  =  3  + 6*'  =  3(1  +  2/). 

Then  m  =  3,  a  =15,  c  =  3. 

The  following  45  integers  constitute  a  complete  residue  system, 
mod  3  +  6V, 

01  234  5  67 

i    1  +  i      2  +  i      3  + »      4  + 1      5  +  t      6  +  *      7  -f « 

2f      1+2*     2  +  2*      3  +  2*     4  +  2*      5  +  2*      6  +  2J     7  +  2* 

8  9  io  ii  12  13  14 

8  +  *      9  +  *      10 +  *      11 +*      12 +  *      13  +  *      14  +  *. 
8  +  2*    9  +  2.1    10  +  2*    11+  2*    12  +  2/    13  +  2*     14  +  2*. 

We  can  thus  obtain  a  complete  residue  system  with  respect  to  any 
modulus  by  means  of  the  method  employed  in  the  above  theorem. 

There  are  two  important  special  cases  which  deserve  mention. 

i.  If  fi  =  p  +  qi,  where  p  and  q  have  no  common  divisor,  the 


u  +  vi,  i 


THE   REALM    k(i).  1 85 

integers  i,  2,  •••,  p2  +  q2,  =  n(n),  form  a  complete  residue  sys- 
tem, mod  fi. 

ii.  //  ix  =  m,  a  rational  integer,  the  m2  integers 

-      I  y  =  o,  1,  •-.,  \m\  —  1, 

form  a  complete  residue  system,  mod  m. 

Ex.  1.     Prove  i  and  ii  without  making  use  of  Th.  8. 

Ex.  2.     Show  that  a  as  13,  mod  7,  implies  a'  ^  /?,  mod  7'. 

All  integers  belonging  to  the  same  residue  class,  mod  /*,  have 
with  fx  the  same  greatest  common  divisors ;  for  from 

a  =  (3,  mod  fi, 

it  follows  that  a  =  fi-\-vfi, 

and  hence  every  common  divisor  of  (3  and  fi  is  also  a  divisor  of  a 
and  every  common  divisor  of  a  and  ^  is  a  divisor  of  (3. 

In  particular,  if  one  number  of  a  residue  class  be  prime  to  the 
modulus,  fi,  all  other  numbers  of  the  class  are  prime  to  pu 

A  system  of  integers  incongruent  each  to  each  with  respect  to 
a  given  modulus,  /x,  and  prime  to  ^  is  called  a  reduced  system  of 
incongruent  numbers,  mod  /a,  or  a  reduced  residue  system,  mod  /*. 
Thus  the  numbers  1,  2,  3,  4,  5,  6,  7,  8,  9,  10  constitute  a  complete 
system  of  incongruent  numbers,  mod  1  +3*,  and  1,  3,  7,  9  con- 
stitute a  reduced  system  to  the  same  modulus. 

§  12.    The  ^-Function  in  k(i). 

Just  as  in  R,  we  understand  by  <£(/*)>  where  ^  is  an  integer  of 
k(i),  the  number  of  integers  in  a  reduced  residue  system,  mod  p. 

We  have  0(c)  =  1, 

where  c  is  any  unit  of  k(i),  and,  as  may  be  easily  seen, 

<f>(Tr)=n[Tr]  —  1, 

where  tt  is  a  prime  of  k(i)  ;  for  example,  <f>(2  -f-f)  =4,  since 
1,  2,  3,  4,  5  constitute  a  complete  residue  system,  mod  2  -f-  i,  and 
all  these  integers  except  5  are  prime  to  2  -f-  i.     Likewise 

<Ki+3*')=4> 


I  86  THE   REALM    k(i). 

since  I,  2,  3,  4,  5,  6,  7,  8,  9,  10  constitute  a  complete  residue  sys- 
tem, mod  1  -\-  $i,  and  of  these  integers  only  1,  3,  7  and  9  are 
prime  to  1  +  3*. 

To  get  a  general  expression  for  <j>(fi)  in  terms  of  n,  we  may 
employ  any  one  of  the  three  methods  used  to  obtain  the  corre- 
sponding expression  in  R. 

We  shall  sketch  the  proof  briefly,  following  the  third  method 
used  in  R  (see  Chap.  Ill,  §4). 

The  completion  of  this  and  the  two  remaining  proofs  will  serve 
as  exercises. 

Theorem  9.  //  a  =  /3y,  where  /?  and  y  are  any  integers  of 
k(i),  there  are  in  a  complete  residue  system,  mod  a,  exactly  w(y) 
numbers  that  are  divisible  by  /?. 

Let  Yi>y2>  •••>yn(V)  1) 

be  a  complete  system  of  incongruent  numbers,  mod  y.  The  num- 
bers Pyi,Py**~->Py*to)  2) 
are  incongruent,  mod  a,  for  if 

/tyfc  =  /tyi>  mod  a, 
then  yfc==y4,  mody, 

which  is  impossible. 

Moreover,  every  integer  (38,  divisible  by  /?  is  congruent  to  some 
one  of  the  numbers  2),  mod  a;  for  8  is  congruent  to  some  one, 
say  yi,  of  the  numbers  i),  mod  y,  and  from 

h^yi,  mody, 

it  follows  that  £8  =  /^,  mod  a. 

Since,  also,  every  integer  congruent,  mod  a,  to  one  of  the  num- 
bers 2)  is  divisible  by  /?  (see  §  n  and  Chap.  Ill,  §  1,  ix),  and  the 
numbers  2)  are  n(y)  in  number,  there  are  in  every  complete 
residue  system,  mod  a,  exactly  «(y)  numbers  that  are  divisible 

by/?. 
Theorem  10.    //  tt  be  any  prime  of  k(i), 


^(Tm)=wM(I__L_) 


THE   REALM    k(i).  I  87 

From  the  last  theorem  we  see  that  among  the  n[irm]  numbers  of 
a  complete  residue  system,  mod  irm,  there  are  exactly  ^[tt"1-1]  that 
are  divisible  by  tt,  and  hence  n[irm]  — n[7rwl~1]  that  are  prime  to 

ttw;  that  is  <£Om)  =w[7rm](  I  —  -ir—  V 

\  »M  / 

To  derive  the  general  expression  for  </>(/*)  we  have  now  to 
prove  the  theorem  for  k(i)  corresponding  to  Th.  4,  Chap.  III. 

Theorem  ii.  //  /ilt  /*,,  •  •  -,fx8  be  integers  of  k(i)  prime  each  to 
each  <£Oi/*2  •••/a«)  =^(^1)^(^2)  •••<£0*«)- 

Ex.     We  have  — 3  +  n«=  (1  +  30  (3  +2t)f 

where  1  +  3*  and  3  -{-  2/  are  prime  to  each  other. 

Hence        <t>{—  3+  lit)  =0(1  +  300(3  +  20  =4-  12  =  48. 

The  proof  of  this  theorem  depends  directly  upon  the  following 
theorem  which  can  be  proved  exactly  as  in  R  (Chap.  Ill,  §  14)  : 

Theorem  12.    //  /x  =  ^1/x2---^« 

where  px,ix2,  '-,n8  are  integers  of  k(i)  prime  each  to  each,  and 
if  a1}a2,  ••■,a8  be  any  integers  of  k(i),  there  exist  integers,  <o, 
such  that 

co  ae  clx,  mod  fix,  to  ^  ol2,  mod  fx2,  •  •  • ,  w  ^  as,  mod  /as, 
and  all  these  integers  are  congruent  each  to  each,  mod  p.     More- 
over o>  =  a1^1  +  a2/32  +  •  •  •  +  a8p8,  mod  ft, 
where 

Pi  =  1,  mod /xj,  and  /?*  ==0,  mod ^  •  •  •  i*>i-1fj>i+1  •  -  •  /**,  *=  1,2,  •  •  •,  ^. 

We  can  now  obtain  easily  the  general  expression  for  <f>([i),  n 
being  any  integer  of  k(i). 

Theorem  13.  If  p  be  any  integer  of  k(i)  a)td  7^, 7r2,  —  ',tt8  the 
different  prime  factors  of  jx,  then 

fG0*=»M  (i-n^)(l-TLTV--(i  --tM- 

Let 


1 88  THE   REALM    k(i). 

By  Th.  ii  we  have 

*00  =  <*>Oiei)4>U2e2)  •••♦(«/»), 

from  which  by  Th.  10  it  follows  that 

+<*>  =n[^,]  ( '  ~spb )  wM  ( ' "  sib) 

-^'](I-)T[i7]) 

and  hence  that 

Ex.  We  have 

—  201  —  43*'  =  (i  +  0  (2  +  i)*(3  +  2O2, 
and  hence  <t>  (—  201  —  43*) 

=»(_M_^(I__^)(,__iR)...(,__f_J:^)> 

=  42250  •  I  •  f  •  H , 
=  15600. 

Theorem  14.    //  B19  82,  •  •  ♦,  8r  fo  ffo  different  divisors  of  n,  then 

1,  r 

For  proof  see  corresponding  theorem  in  R  (Chap.  Ill,  Th.  6). 

Ex.    We  have     —  3  +  ni  =  (1  +0  (2  +  •)  (3  -f  2.1). 

The  different  divisors  of  — 3  +  11*  are  1,  1  + 1,  2  +  t,  3  +  2^  *  +  3*> 
i  +  5*>  4  +  7**  and  — 3  +  11*,  and  for  these  the  corresponding  values  of 
0  are  1,  1,  4,  12,  4,  12,  48,  48,  whose  sum  is  seen  to  be  130,  =  n[ —  3  +  111]. 

§  13.  Residue  Systems  Formed  by  Multiplying  the  Numbers 
of  a  Given  System  by  an  Integer  Prime  to  the  Modulus. 

Theorem  15.  //  j^,/^,  •••,**»[*]  be  a  complete  residue  system, 
mod  fi,  and  a  any  integer  prime  to  /*,  then  a^, a/x2,  •  •*,a/*n[M] 
is  also  a  complete  residue  system,  mod  p. 

The  integers  a^,  a/x2,  •  •  • ,  a/*„[>]  are  incongruent  each  to  each, 
modju,  for  from 

OLfii^dixj,  mod//., 


THE   REALM    k(i).  1 89 

it  would  follow  that,  since  a  is  prime  to  p, 

fii^fjij,  mod  jx, 

which  is  contrary  to  the  hypothesis  that  fa,  p2,  •  •  • ,  pn[^  form  a 
complete  residue  system,  mod  p.  The  integers  a/x,15  ap2,  •  •  •,  apn[^ 
are,  moreover,  n[fi]  in  number.  They  form,  therefore,  a  complete 
residue  system,  mod  p. 

Cor.  //  plt  p2,  — ' ,  p4>(m)  be  a  reduced  residue  system,  mod  p., 
and  a  be  prime  to  p,  then  ap1}ap2,  •  -  • ,  ap^^)  is  also  a  reduced 
residue  system,  mod  p.;  for  aplyap2,  •  '*,ajp+(»)  are  incongruent 
each  to  each,  mod  ^,  prime  to  p  and  <£(/x)  m  number. 

§14.    The  Analogue  for  k(i)  of  Fer mat's  Theorem. 
A  theorem  analogous  to  the  generalized  Fermat's  theorem  for 
rational  integers  can  be  proved  for  the  integers  of  k(i)  ;  that  is, 

Theorem  16.  //  p  be  any  integer  of  k(i)  and  a  any  integer 
prime  to  p,  then  a***3  =  i,  mod^. 

Let  alya2,  •••,a</>(tt)  be  a  reduced  residue  system,  mod  p.;  then 
aalfaa2,  •••,aa^(M) 
is  also  such  a  system  (Th.  15,  Cor.). 

Since  aax,aa2,  ••-,aa^(|1) 

and  alta2i  •••,a^(M) 

are  both  systems  of  this  kind,  each  integer  in  the  one  system  must 
be  congruent,  mod  p.,  to  one  and  only  one  integer  in  the  other  sys- 
tem, though  perhaps  in  a  different  order;  that  is, 

aa2       =  ak 


aa<tt^)=ak<i)(ilx)  . 


mod  p. 


Hence 


a*Maxa2--  a*(M)  z=aklaki  •••  afc^(|0,  mod/*, 
and  since  axa2  •  •  •  a$(M)  =  aklaki-  - •  afc</>(M), 


190  THE   REALM    k(i). 

and  is  prime  to  /a,  we  have 

a*(n)===Ij  mod/*. 
Cor.  1.    If  -rr  be  a  prime  and  a  any  integer  not  divisible  by  it, 
then  a»[T]-i  ==  lf  m0(j  w 

This  is  the  analogue  of  Fermat's  Theorem. 

Cor.  2.     If  tt  be  a  prime  and  a  any  integer  of  k(i),  then 

ara[flrl==a,  modTr. 

Ex.   1.    Let  7r=i  +  2«,   and   a=i  +  i; 

then  (1  +  Q+CM-iOpBi  1,  mod  1  +  2f, 

or  (i-\-i)*  =  —  4=1,  mod  1 +2*. 

Ex.  2.     Let  fi=  1  +  31  and  a  =  3 ; 

then  3<*>(i+3i)^  Xj  mo(j  j  _j_  £ 

or  81  ^  1,  mod  1  +  31. 

Ex.  3.  If  a  and  /*  be  any  two  integers  of  k{%)  and  a  =  cci8,  n  =  f^S, 
where  8  is  the  greatest  common  divisor  of  a  and  /*,  show  that  the  necessary 
and  sufficient  condition  for 

a+GO+i^o,  mod  /*, 

is  that  nx  be  prime  to  5. 

§  15.    Congruences  of  Condition. 

The  remarks  at  the  beginning  of  §9,  Chap.  Ill,  apply  equally 
to  congruences  in  k{%)9  and  the  theory  of  congruences  of  con- 
dition in  k(i)  can  be  developed  in  exactly  the  same  manner 
as  in  R. 

In  k(i)  the  coefficients  of  the  polynomials  are  any  integers 
of  k(i). 

With  this  change  we  can  show  that  a  polynomial  in  a  single 
variable  x  can  be  resolved  in  one  and  but  one  way  into  prime 
factors  with  respect  to  a  modulus  which  is  a  prime  of  k(i),  and 
upon  this  theorem  build  a  theory  for  congruences  in  one  unknown 
just  as  in  R. 

The  theories  of  power  residues,  binomial  congruences  and  in- 
dices may  be  developed  similarly  for  the  integers  of  k{t). 


THE   REALM    k(i).  I9I 

§  16.     Two  Problems. 

We  shall  now  discuss  briefly  two  problems  which  are  of  interest 
in  the  theory  of  numbers,  the  first  being  especially  famous.  They 
can  be  solved  without  making  use  of  numbers  other  than  those 
of  R,  but  their  solution  is  greatly  assisted  by  the  introduction 
of  the  realm  k(i). 

Problem  i.  To  represent  a  rational  prime  as  the  sum  of  two 
squares.1 — Let  p  be  a  rational  prime  and  suppose  the  desired  rep- 
resentation possible.     Then 

p  =  a2  +  b2, 
and  hence  p  =  (a  +  bi)  (a  —  bi)  ; 

that  is,  the  representation  is  possible  when  and  only  when  p  is  the 
product  of  two  conjugate  primes  of  k(i).     Hence 

i.  No  prime  of  the  form  411  -f-  3  can  be  represented  as  the  sum 
of  two  squares,  since  a  prime  of  this  form  is  a  prime  in  k(i). 

ii.  The  number  2  and  every  prime  of  the  form  411  -{- 1  can  be 
represented  as  the  sum  of  two  squares. 

Moreover,  this  representation  is  unique,  for  if  we  have  two  dif- 
ferent representations 

p  =  a2  +  b2  and  p  =  a1*  +  b12, 
then 

p=(a-\-bi)(a —  bi)  and  p=  (ax  +  bj)  (ax  —  bj)  ; 

that  is,  p  would  be  factorable  in  two  different  ways  into  prime 
factors  in  k(i),  which  is  impossible.  Hence  2  and  every  prime 
of  the  form  qn  -f-  1  can  be  represented  in  one  and  only  one  way  as 
the  sum  of  two  squares,  but  no  prime  of  the  form  411  -f-  3  can  be 
so  represented. 

Problem  2.  To  represent  any  positive  rational  integer,  m, 
as  the  sum  of  two  squares. 

Let  m  =  pxp2  •  •  •  pr-qi*^**  -  •  •  qJ; 

where  pltp2,  '--,pr  are  rational  primes  of  the  form  411  -j-  1  or  2, 

1Fermat:  Works,  Vol.  I,  p.  294. 

For  solution  of  this  problem  without  the  aid  of  k(i)  see  Dirichlet- 
Dedekind:  §68;  also  Mathews:  §91. 


192  THE   REALM    k(i). 

two  or  more  of  which  may  be  alike,  and  qltq2,m",q8  rational 
primes  of  the  form  4W  +  3,  that  are  all  different  from  one  another. 
If  the  representation  be  possible, 

m  =  a2  -f-  b2 ; 

and  hence  m  =  (a  -f-  bi)  (a  —  bi) . 

The  representation  is  therefore  possible  when  and  only  when  we 
can  factor  m  into  two  conjugate  factors  in  k(i).  The  necessary 
and  sufficient  condition  for  this  is  that  all  the  fs  be  even,  in  which 
case  we  have,  if 

Pi  =:=  1T\K\  ,    P2  =  7T27r2  ,  '  ",pr  =  TTrTTr  , 

m  =  (»!»,  •  •  •  atfe*^,"*  ■  ■  ■  qsu'*)  X 

Hence  if  a  positive  rational  integer,  m,  contain  a  prime  factor 
of  the  form  4n-\-  3  an  odd  number  of  times,  m  cannot  be  repre- 
sented as  the  sum  of  two  squares.  In  all  other  cases  the  repre- 
sentation is  possible. 

Moreover,  supposing  the  factorization  1)  to  be  possible,  it  can 
be  effected  in  general  in  several  different  ways,  as  for  example, 

m  =  (»>,  . . .  wrtflfiqf*  •  •  •  q.u'2)  X 

and  since  each  of  these  factorizations  yields  a  different  represen- 
tation of  m  as  the  sum  of  two  squares,  the  problem  can  be  solved 
in  exactly  as  many  different  ways. 

If  m~2npxeip2e*-'prerq1Hq2t2'--qSiu,  where  the  p's  are  primes 
of  the  form  4n  -\-  1,  all  different,  the  q's  primes  of  the  form 
4n  +  3>  and  the  ?s  oil  even,  then,  if  N  be  the  number  of  different 
ways  in  which  m  can  be  represented  as  the  sum  of  two  squares, 
we  have  N  =  i(e1  +  1)  (e2  +  1)  -  •  •  (er  +  i)  or\(ex  +  1)  (e2  +  1) 
•  *  •  (er  +  1)  +  i  according  as  some  or  none  of  the  e's  are  uneven. 
(See  Gauss:  Disq.  Arith.,  V,  182.) 

Ex.         65  =  13  •  5  =  (1  +  2O  (1  —  »)  (2  +  3O  (2  —  31), 

=  [(i+20  (2 +  301  [0 -20(2-30], 
=  (-4  +  70(-4-70=42  +  72, 

or  as  [(i  +  20(2  —  30H(i— 20(2  +  30], 

=  (8  +  0(8-0  =82  +  i2. 

Thus  65  can  be  expressed  in  two  ways  as  the  sum  of  two  squares. 


THE   REALM    k(l).  1 93 

§  17.    Primary  Integers  of  k(i). 

When  an  integer,  a,  plays  the  role  of  divisor  it  is  unnecessary 
to  distinguish  between  its  associates.  This  is,  however,  not  the 
case  when  a  is  combined  with  other  numbers  by  the  operations 
of  addition  or  subtraction.  For  example,  when  a  is  the  modulus 
of  a  congruence  we  may  consider  a  to  be  any  one  of  its  asso- 
ciates, but  when  a  is  a  coefficient  some  particular  one  of  its  asso- 
ciates must  be  designated.  This  distinction  between  the  associates 
of  a  is  the  same  as  that  made  in  the  rational  realm  between  a 
and  — a. 

There,  for  example,  the  quadratic  reciprocity  law  is  given  for 
positive  primes,  since  although  we  have  always 

(0-&> 

we  do  not  have  in  general 


is)-m 


An  integer  so  singled  out  from  its  associates  according  to  some 
prescribed  rule  is  called  a  primary  integer. 

This  rule  of  selection  should  evidently  be  such  that  the  product 
of  any  two  primary  integers  is  primary;  that  is,  if  a  and  p  be 
the  integers  selected  as  primary  from  a,  —  a,ia,  —  ia- and  p,  — P, 
ip, —  ip,  respectively,  then  ap  should  be  the  integer  that  ac- 
cording to  the  same  rule  should  be  selected  as  primary  from 
ap,  —  ap,iap,  —  iap. 

Gauss  gives  two  rules  of  selection,  both  of  which  obey  the 
principle  just  enunciated.  The  first  rule  is  based  entirely  upon 
this  principle,  the  second  partially.  Gauss  makes  use  of  the  sec- 
ond rule  and  this  one  will  now  be  described. 

The  rule  will  be  given  here  without  employing  the  above  men- 
tioned principle,  and  will  then  be  shown  to  obey  it. 

We  first  divide  the  integers  of  k(i)  into  two  classes  according 
as  their  norms  are  odd  or  even,  those  of  the  first  class  being  called 
odd  integers,  those  of  the  second  class  even  integers.1 

1  Bachmann :  Die  Lehre  von  der  Kreisteilung,  p.  152. 
13 


194  THE  REALM   k(i). 

If  n[a-\-  bi],  =  a2  +  b2,  be  odd,  it  is  evident  that  either  a  or  b 
is  odd,  the  other  even. 

If  n[a  +  bi]  be  even,  a  and  &  are  both  odd  or  both  even. 

Every  prime  of  k(i)  except  I  +  i  is  evidently  an  odd  integer. 
Since  I  -\-i  and  I — i  are  associates,  it  is  evident  that  n[a]  divis- 
ible by  2  is  not  only  a  necessary  but  a  sufficient  condition  that  a 
shall  be  divisible  by  I  -f-  i. 

We  see,  therefore,  that  a  necessary  and  sufficient  condition  for 
an  integer  of  k(i)  to  be  even  is  that  it  shall  be  divisible  by  i  -f-  i- 

The  selection  of  one  of  the  four  associates  of  an  integer  is  now 
made  as  follows.  Considering  first  only  the  odd  integers  of  k(i), 
we  have  the  following  rule: 

That  number  x  -\-  yi  of  the  four  associated  odd  integers 

a  -\-  bi,  —  a  —  bi,  —  b  +  ai,  b  —  ai  i ) 

is  singled  out  as  primary  in  which  we  have  simultaneously  either 

x*s~       i ;  y  =  o  ^ 

V  ,  mod 4,  2) 

or  x  =  —  i ;  y  =  2  ) 

where  x  denotes  the  real  part  and  y  the  coefficient  of  i. 

That  one  and  only  one  such  integer  exists  in  the  group  i)  is 
shown  as  follows.  Since  a  -\-  bi  is  an  odd  integer,  a  and  b  can 
neither  be  both  odd  nor  both  even.  Suppose  a  even,  b  odd. 
Then  one  of  the  integers,  b  or  — b,  is  of  the  form  4»  +  i,  the 
other  of  the  form  ^n —  i. 

If  now  a  =  o,  mod  4, 

that  one  of  two  integers,  b  —  ai,  —  b  -j-  ai,  will  be  primary  in 
which  the  real  part  has  the  form  411  -f-  1  • 

If  a  =  2,  mod  4, 

that  one  of  the  integers,  b  —  ai,  —  b  -f-  ai,  will  be  primary  in 
which  the  real  part  has  the  form  4^ —  1. 

It  is  evident  in  both  these  cases  that  none  of  the  remaining 
associates  satisfy  the  conditions. 

Similarly  we  see  that  when  a  is  odd  and  b  even,  one  and  only 
one  of  the  four  associates  1)  satisfies  2). 


THE   REALM    k(i).  1 95 

If  a  be  a  rational  integer,  that  one  of  the  integers,  a,  —  a,  is 
primary  which  has  the  form  411  +  1 .  The  negative  rational 
primes  prime  in  k(i)  are  thus  seen  to  be  primary.  Two  conju- 
gate odd  integers  are  evidently  either  both  primary  or  both  non- 
primary.  It  can  be  easily  shown  that  the  above  rule  of  selection 
is  equivalent  to  the  following : 

That  one  of  four  associated  odd  integers  is  primary  which  is 
congruent  to  1,  mod  2  +  21.  3) 

Ex.     Of  the  four  associated  odd  integers 

9  -f- 121,  —  9  —  12*,  12  —  gi,  —  12  -f-  gi, 
9 +121  satisfies  the  conditions  2);   for  we  have 
9^1  and  12^0,  mod 4. 

Hence  9  +  12*  is  primary. 

We  also  see  that  9  +  12*  =  1,  mod  2  -\-  22. 

It  is  easily  seen  that  9  -f- 12*  is  the  only  one  of  its  associates  which 
satisfies  the  conditions  2)    or  their  equivalent  3). 

Since  every  prime  of  k(i)  except  I  +  *  is  an  odd  integer,  we 
can  now  distinguish  between  the  associates  of  every  prime  except 
1  -}-  *.  In  the  case  of  1  +  i  we  may  take  any  one  of  its  associates, 
say  1  +  i  as  the  primary  one.  The  primary  primes  of  k  (t ) 
whose  norms  are  less  than  50  are 

I  +t,  —  1+21,  —I—2i,  —3,   3  +  2/,   3  —  21,    I  +4*,    I—4/, 

—  5  +  2«,  —  5  —  2/,  —  1  -f  6i,  —  1  —  6i,  5  +  4h  5—4*',  — 7- 

Remembering  that  a  necessary  as  well  as  sufficient  condition 
for  an  integer,  fx,  to  be  even  is  that  it  shall  be  divisible  by  1  +  i, 
we  can  distinguish  between  the  associates  of  /a  by  taking  that  one 
as  primary  which  when  written  in  the  form  ( 1  -\-i)  nv  has  the 
factor  v,  which  is  an  odd  integer,  primary.  We  shall  now  show 
that  the  product  of  two  odd  primary  integers  is  a  primary  integer. 
Let  a,  = a  -j-  hi,  and  j3,  =  c  +  di,  be  any  two  odd  primary  in- 
tegers.    Then  one  of  the  following  cases  must  occur. 


mod  4, 


1. 

a=  1 

11. 

1 

1 
a==- 

11. 
—  1 

IV. 

ass—  1  * 

b  =  o 

b  = 

0 

b  = 

2 

b==        2 

c==  1 

c  =  - 

- 1 

c^ 

1 

C  = I 

d  =  o 

d  = 

2 

d^ 

0 

dz=      2  J 

I96  THE   REALM    k(i). 

and  afi=(ac —  bd)  -f-  (ad -\-  bc)i  =  e  +  fi, 

gives  one  of  the  following  corresponding  cases : 
i.  ii.  iii.  iv. 


V  ,  mod  4. 

Mi' 


/  =  0  f&B         2  /eee         2 

Hence  a($  is  always  an  odd  primary  integer,  if  a  and  /?  be  odd 
primary  integers.  This  may  be  shown  more  simply  by  means  of 
the  condition  3). 

From  this  it  follows  at  once  that  the  product  of  any  two  pri- 
mary integers  is  primary.  We  may  now  express  the  unique  fac- 
torization law  for  the  integers  of  k(i)  as  follows: 

A  primary  integer  can  be  resolved  in  one  and  only  one  way  into 
a  product  of  primary  prime  factors. 

The  term  primary  integer  is  generally  taken  to  mean  what  is 
here  called  an  odd  primary  integer. 

§  18.  Quadratic  Residues  and  the  Quadratic  Reciprocity  Law 
in  k(i).1 

If  a  and  fx  be  any  integers  of  k(i)  prime  to  each  other,  we  say, 
as  in  R,  that  a  is  a  quadratic  residue  or  non-residue  of  /u.  accord- 
ing as  the  congruence 

x2  =  a,  mod  fi, 
has  or  has  not  roots. 

Ex.  1.     The  congruence 

x*  as  1  -f-  i,  mod  1  —  21, 
has  the  roots  ±  2 ;  for 

(±  2)2  =  1  +  i,  mod  1  —  2.1, 

since  4  —  ( 1  +  *)  =3  —  i=  ( 1  -j-  *)  ( 1  —  21) . 

Hence  1  -f-  i  is  a  quadratic  residue  of  1  —  2fc 
Ex.  2.    On  the  other  hand  the  congruence 

x2  s  3,  mod  1  —  2%, 

has  no  roots,  for  substituting  the  integers  ±  I,  ±  2,  of  a  reduced  residue 
system,  mod  1  —  2%,  we  have 

\  ,  mod  1  —  2i. 
4^3  j 

xSee  Gauss:  Theoria  Residuorum  Biquadraticorum,  §§  56-60;  Works. 
Vol.  2,  pp.  126-130. 


THE   REALM    k(i).  \  gf 

Hence  3  is  a  quadratic  non-residue  of  I  —  2f. 

The  theory  of  quadratic  residues  can  be  developed  for  k(i) 
along  lines  so  nearly  identical  with  those  for  the  same  subject  in 
the  rational  realm  that  only  the  briefest  outline  will  be  given  here. 

We  have,  as  before,  two  questions  to  answer:  first,  what  in- 
tegers are,  and  what  are  not,  quadratic  residues  of  a  given  modu- 
lus; second,  of  what  moduli  is  a  given  integer  a  quadratic  residue 
and  of  zvhat  moduli  is  it  a  non-residue ? 

The  first  question  can  be  easily  answered.  The  second  is  much 
more  difficult.  We  shall  confine  ourselves  in  what  follows  to  the  case 
where  the  modulus  is  a  prime  r.  We  observe  first  that  every  odd 
integer  of  k(i),  that  is,  every  integer  prime  to  I  +  h  is  congruent 
to  I,  mod  I  +  h  and  hence  is  a  quadratic  residue  of  I  -f-  i. 

For  7r,  an  odd  prime,  we  have  the  following  theorem,  the  proof 
of  which  is  like  that  of  the  corresponding  theorem  for  rational 
integers  (Chap.  IV,  Th.  i). 

Theorem  17.  The  necessary  and  sufficient  condition  that  a 
shall  be  a  quadratic  residue  of  ir  is  that 


Ex.  3.    Let  tt=  1  —  2.x,  a=  1  +  i.    We  have 

n|>]-l 

(1  +  0     2      =  (1  +  02  =  2t  as  1,  mod  1  — 2i. 
Hence  1  -f- 1  is  a  quadratic  residue  of  1  —  21. 
Ex.  4.    Let  7T  ss  1  —  21,  a  =  3.    We  have 

»[ir]-l 

3     2      =32  =  9^i,  mod  1—  2»* 

Hence  3  is  a  quadratic  non-residue  of   1  —  21.     These  results  are  con- 
firmed by  Ex.'s  1  and  2  above. 

Cor.     The  integer  a  is  a  quadratic  residue  or  non-residue  of  ir 
according  as  we  have 

a     2    =1  or  — 1,  modir.1 
Let  now,  as  in  the  rational  realm,  the  symbol  (a/n)  have  the 
1  See  Chap.  IV.  Th.  1.  Cor.  1. 


I98  THE   REALM    k(i). 

value  1  or  —  1  according  as  a  is  a  quadratic  residue  or  non-residue 
of  sr,  we  have 

I  -  )  m  a     2     ,  mod  it. 

The  symbol  (a/V)  obeys  the  following  laws 
i.  If  «  =  /?,  modTr, 

te)-(f-)' 

a  Sine.  (=-!)_  ('-)    _,, 

--    (v)-(t)  (:-)=(,-)■ 


iv.  Since 


(l)-(^)--(;)-(i) 


it  follows  that 


©-(-«)-(i)-(-:). 


v.  Since  y2  =  a,  modir, 

implies  y'  ^=ol',  mod  7/, 

we  have 


i)-6) 


Every  integer  a  can  be  written  in  the  form 

a  =  ir(i+i)8plP2--Pn, 

where  r  =  o,  1 ,  2  or  3,  s  =  o  or  a  positive  integer,  and  px,  p2>  *  *  •  >  P* 
are  odd  primary  primes.     We  have  then 

(=)-a)-(^m)(?)-fe), 

and  the  determination  of  the  value  of  (a/v)  is  seen  to  be  resolved 
into-  the  determination  of  the  values  of 


THE   REALM    k(i).  1 99 


aM^'MD 


where  p  is  an  odd  primary  prime. 

The  close  similarity  between  this  resolution  of  our  original 
problem  into  simpler  ones  and  the  corresponding  case  in  the 
rational  realm  should  be  noticed. 

Theorem  18.  The  unit  i  is  a  quadratic  residue  or  non-residue 
of  a  prime  it  according  as  n[?r]  is  of  the  form  8m  -\-  i  or  8  m  +  5- 

If  ir  be  a  prime  of  the  first  degree,  w[tt]  is  a  positive  rational 
prime  of  the  form  4k  +  1,  and  hence  either  of  the  form  8w  +  1 
or  8m  +  5. 

If  7T  be  a  prime  of  the  second  degree,  n[ir]  is  the  square  of  a 
rational  prime  of  the  form  4k  +  3,  and  hence  of  the  form  8m  -|-  1. 

We  have  from  Th.  17 

2     ,  mod  7T, 


u)- 


(i\  »r»]-i 

and  hence  (-)==(—  1)      4     ,  mod  ir, 

»[ir]-l 

or  since  (_  ,j     4     =  T  or  _  z 


(';)-<-> 


»[*]-! 
4 


But  («[ir]  — 1)/4  is  even  or  odd  according  as  n[ir]  is  of  the 
form  8m  +  1  or  8m  +  5. 

Hence  (*/*•)  =  1  or  —  1  according  as  n[ir]  is  of  the  form 
8m  +  1  or  8m  +  5-  We  observe  that  i  is  a  quadratic  residue  of 
all  primes  of  the  second  degree.  The  solution  of  the  same  ques- 
tion for  1  +  i  is  obtained  by  Gauss  inductively  as  follows  i1 

We  find  by  means  of  Th.  17  that  1  +  i ^  is  a  quadratic  residue 
of   the   following   primary   primes   — I  +  21,    3  —  2f,   — 5 — 21, 

—  1—  6/,   5+4*',   5  —  4*',    —7,    7  +  2',   —  5  +  &,    etc.,    and    a 
quadratic  non-residue  of  — 1 — 21,  — 3,  $-{-2i,   I +4*,   1 — 4*, 

—  5  +  21,-1+61,  7  —  2/,  —5  —  6/,  —3  +  81',  —3  —  8/,  5  +  8/, 
5  —  8/,  9  +  4*,  9  —  4/,  etc. 

*Th.  Res.  Biquad.,  Com.  Sec,  §58;  Works,  Vol.  II,  p.  128. 


200  THE   REALM    &(*). 

Examining  these  series  of  primes  we  see  that  those  in  the  first 
class  are  all  such  that 

a-\-  &=  i,  mod  8, 
and  those  in  the  second  class  such  that 

a  +  b  bb  —  3,  mod  8. 

Hence  it  seems  probable  that  I  + ' 'is  a  quadratic  residue  or  non- 
residue  of  an  odd  primary  prime,  a  -\-  hi,  according  as  we  have 

a  +  b  =  i  or  —  3,  mod  8, 

one  of  which  cases  must  always  occur  (see  definition  of  primary 
integer). 

Since  the  quadratic  character  of  an  integer  is  the  same  with 
respect  to  all  associates  of  tt,  and  in  particular 


\a  +  bij       \  —  a  —  biJ 


we  see  that,  if  the  above  induction  be  correct,  I  +  *  is  a  quadratic 
residue  or  non-residue  of  — a  —  bi  according  as 

—  a  —  &  =  —  I  or  3,  mod  8, 

a  +  bi  being  an  odd  primary  prime. 

Assuming  the  correctness  of  the  above  inductive  reasoning,  we 
have  the  following  theorem : 

Theorem  19.  If  a-\-  bi  be  a  prime  such  that  a  is  odd  and  b 
even,  1  -\-i  is  a  quadratic  residue  or  non-residue  of  a-\-  bi,  ac- 
cording as  a-\-b=±  1  or  ±  3,  mod  8. 

This  theorem  may  be  proved  by  treating  it  as  a  special  case  of 
a  more  general  theorem  (Th.  22),  which  we  shall  consider  in  the 

next  section.1    To  determine  the  value  of  (         .  .]  we  have  only 

to  remember  that 

(^•)=(S)(seevabove)- 

1  For  an  independent  proof  see  Dirichlet,  Crelle,  Vol.  XXX,  p.  312. 


THE   REALM    k(i).  201 

and  hence  since 

(  j. )     =  i,whena+( — b)  as  ±  i,  modSt 

and  =  —  i,  when  a  +  ( —  &)  ■■  dz  3,  mod  8, 

we  have 


V*  +  &7 


=  1,  when  a —  &  =  ±  1,  mod  8, 
: — 1,  when  a  —  &=±3,  mod  8. 

Ex.  1.    Deduce  the  above  criterion   for  the  value  of    (  — ^—  )   from 

Va-f-fr*/ 

the  fact  that  (l=*\  =  (     *A  f-l±J^  . 

Ex.  2.     Under  what  condition  is 


\a  +  bi)       \a  +  bij 


Gauss  proceeds  next  to  the  consideration  of  the  question:  Of 
what  odd  primary  prime  moduli  is  a  given  odd  primary  prime  a 
quadratic  residue  and  of  what  a  non-residue?  The  analysis  em- 
ployed in  the  discussion  of  this  question  so  beautifully  exemplifies 
what  can  be  accomplished  in  the  theory  of  numbers  by  induction, 
this  constituting,  as  Gauss  says,1  "  the  peculiar  charm  "  of  this 
branch  of  mathematics,  that  we  shall  give  it  in  full.    , 

The  following  is  a  free  translation  of  §§  59,  60,  Commentatio 
Secunda,  Theoria  Residuorum  Biquadraticorum. 

"  Passing  to  the  odd  prime  numbers,  we  find  the  number  —  1  +  21 
to  be  a  quadratic  residue  of  the  moduli  3  -f-  2i,  1  —  4*,  —  5  -\-  21, 

—  i—6i,  7  —  2i,  _3  +  8i,  5  +  8*,  5  —  81,  9  +  4,  etc.,  but  a 
non-residue  of  the  moduli  —  1  —  2i,  —  3, 3  —  2»,  1+4*,  —  1  +  6t, 
5  +  4h  5  —  4*',  —  7>  7  +  2i,  —5  +  6/,  —5  —  61,  —  3  —  81,  9  —  4*, 
etc. 

Reducing  the  moduli  of  the  first  class  to  their  residues  of  least 
absolute  value  with  respect  to  the  modulus  —  1  -f-  2i,  we  find  these 
to  be  —  1  and  1  only ;  for  instance,  3  +  2*  = —  i,  1  — 4**2= —  i, 

—  5  -|-2«=  1,  — 5  —  212s —  1,  etc. 

1  Gauss:  Works,  Vol.  II,  pp.  152  and  157. 


202  THE   REALM    k(i). 

On  the  other  hand,  all  moduli  of  the  second  class  are  found  to 
be   congruent  to   either  i  or  — i  with   respect  to  the  modulus 

—  I  -f-  21.1 

But  the  numbers  i  and  —  i  are  themselves  quadratic  residues 
of  the  modulus  —  I  -f-  2*,  while  i  and  — i  are  non-residues  of  the 
same  modulus ;  wherefore,  so  far  as  induction  may  be  trusted,  we 
obtain  the  theorem :  The  number  —  I  -\-  21  is  a  quadratic  residue 
or  non-residue  of  the  prime  number  a-\-bi  according  as  a  -j-  bi 
is  a  quadratic  residue  or  non-residue  of  —  1  +  2%  itself,  if  a  +  bi 
be  the  primary  one  of  its  four  associates,  or  more  exactly  if  merely 
a  be  odd  and  b  even. 

Moreover,  from  this  theorem  follow  immediately  similar  theo- 
rems for  the  numbers  1  —  2i,  —  1  —  zi,  1  -f-  2.i. 

Since       (L~2i\  =  (  —l  \  (~1  +  2i\  =  (~I+2i\ 

\a  +  bi)       \a  +  bi)\    a  +  bi  )       \   a  +  bi  /' 

we  have  (l=g)  =  (2  +  «Y 

\a-{-biJ       \i — 2%) 

Also  f-1-2^  =  (-1±2\  =  (JlZZ*L\  =  (  °  +  bi\ 

\   a  +  bi  J       \   a  —  bi  )       \— 1+21  J       \—i—2iJ 

and  then  as  above  (*+f\  =  (a  +  hi\ 

\a  +  bi)       \i+2ij' 

Instituting  a  like  inductive   enquiry  concerning  the   numbers 

—  3  or  3,  we  find  that  both  are  quadratic  residues  of  the  moduli 
3  +  2*,  3  —  2f,  —  1  +  6»,  —  1  —  6i,  —  5  +  6i,  —  5  —  &,  —  3  +  Si, 

—  3  —  Si,    9  -f-  4i,    9  —  zji,    etc.,    but   non-residues    of   —  1  -f-  2f, 

—  1—2?;    1+4*,    i—  4*,   —5  +  2i,  —s  —  2i,   5  +  4*',   5—4^ 
7  +  2*,  7  —  2*,  5  +  8*;  5  —  8*,  etc. 

The  former  are  congruent  with  respect  to  the  modulus  3  to 
some  one  of  the  four  numbers  1,  —  1,  i,  — i;  the  latter,  however, 
to  some  one  of  the  four  numbers  1  -\-i,  1  — i,  —  I  +.*,  —  1  — i.2 

1  It  will  be  observed  that  1,  — 1,  i,  — -i  constitute  a  reduced  residue 
system,  mod  —  1  +  2*- 

2  The  numbers  1,  — 1,  i,  — i,  i-\-i,  J  —  i>  — *+**  — 1 — *  constitute 
a  reduced  residue  system,  mod  3. 


THE    REALM    k{l).  2C>3 

The  numbers  i,  — i,  i,  — i  are  themselves  quadratic  residues 
of  3,  while  I  +  i  x  — h  —  I  +  **  —  *  — *  are  non-residues. 

Induction  teaches,  therefore,  that  the  prime  number  a  +  bi, 
supposing  a  odd,  b  even,  has  the  same  relation  to  the  number  —  3 
as  —  3  has  to  a  -f-  hi,  in  so  far  as  the  one  is  a  quadratic  residue  or 
quadratic  non-residue  of  the  other,  and  like  relations  hold  between 
3  and  a  -\-  bi. 

Applying  a  like  inductive  process  to  other  prime  numbers,  we 
find  in  every  case  this  most  elegant  law  of  reciprocity  confirmed, 
and  in  the  arithmetic  of  the  complex  numbers  we  are  led  to  this 
fundamental  theorem  concerning  quadratic  residues : 

Theorem  20.1  //  a1  -f-  °xi  and  a2  -f-  b2i  be  two  prime  numbers 
such  that  ax  and  a^  are  both  odd,  bx  and  b2  both  even,  then  each 
will  be  a  quadratic  residue  or  each  will  be  a  quadratic  non-residue 
of  the  other. 

But  notwithstanding  the  extreme  simplicity  of  the  theorem  its 
demonstration  presents  great  difficulties,  which,  however,  shall 
not  delay  us  here,  since  the  theorem  itself  is  merely  a  special  case 
of  a  more  general  theorem,  which  exhausts,  as  it  were,  the  whole 
theory  of  biquadratic  residues."  We  shall  conclude  this  brief 
resume  of  the  theory  of  quadratic  residues  in  k(i)  with  the  solu- 
tion of  three  examples. 

Ex.  1.     To  determine  the  quadratic  character  of  5  —  4*"  with  respect  to 
the  modulus  11  -\-6i. 
We  have  by  the  above  theorem 

\ii+6i)       \5  —  4*/       V5— 4//' 

1  Since  (  — )=  ( )=  ( )=  ( )  it  is  not  necessary  to  limit  a  and 

*  to  odd  primary  integers,  but  only  to  odd  primary  integers  or  those  with 
their  signs  changed ;  that  is,  integers  of  the  form  a  -f-  bi,  where  a  is  odd 
and  b  even. 

Expressed  symbolically  the  theorem  is 


tax  -f  bti\  _  /ch  -f  b2i\ 
\ch  -f-  b2i)  ~  \tf!  -{-  bii) 


Dirichlet  gives  a  simple  proof  independent  of  the  theory  of  biquadratic 
residues ;  Crelle :  Vol.  IX,  p.  379 ;  also  H.  J.  S.  Smith :  Works,  Vol.  I,  p.  76. 


204  THE   REALM    k(i). 

But  6+ioi=  (i  +  03(i—  4*)- 

Hence  &=£L)-(l±i)\i=gi 

•    \5  — 4V  \5  —  4*/ 

But    f  ^±A^  =  1,  since  5+  (—4)  =1,  mod 8,  (Th.  19),  and  by  Th.  20 
\5  —  4*/ 

\  5  —  4*  /       \  1  —  41  /       \  1  —  4*  /       V 1  —  4l  / 

Hence  (  S~4\)  =  1, 

Vii  +  6t/ 

and  the  congruence  x2  ma  5  —  4/,  mod  11+  6i, 

has  roots. 

Ex.  2.  To  determine  the  prime  moduli  of  which  1  +  2.1  is  a  quad- 
ratic residue,  and  those  of  which  it  is  a  non-residue.  Let  a  -f-  bi  be  a 
primary  prime  and  hence  a  odd,  and  b  even. 

Then 

\a  +  bi)  =  \T+m)  =  VF+2?/'  \T+2i)'  Vj+2? )  or  VT+JwJ' 

according  as   a -\- trims  I,   i,  — 1,   or  — i,  modi-f-2&. 
But 

(*  )  =  T)  (  f\  =_ ,  (^1.)  =  I;  and  (j=L)  =_I( 

\l+2t/  \l+2*/  \I+2Z/  Vl+2*/ 

Hence  1  +  21  is  a  quadratic  residue  of  a  +  fo  when 

a  +  fee  ==  1  or  —  1,  mod  1  +  2*" 
and  a  quadratic  non-residue  when 

a  -j-  fo  ^  *  or  —  iy  mod  1  -f-  2*. 

Therefore  1  -|-  21  is  a  quadratic  residue  of  all  primary  primes   included 

in  the  forms  p(i-\-2i)  ±1,  1) 

and  a  quadratic  non-residue  of  all  primary  primes  included  in  the  forms 

/a(i  +2/)  ±  i.  2) 

Ex.  3.     To  determine  the  prime  moduli  of  which  3  -j-  6*  is  a  quadratic 
residue,  and  those  of  which  it  is  a  non-residue. 
Let  a  -j-  bi  be  a  primary  prime. 

We  have  ft±|ft  =  (— L-}  fl±£\ 

We  find  as  in  the  last  example  that  (      fu) =z  h  w^en  a  +  bi  is  a 


THE   REALM    k(i).  205 

primary  prime  contained  in  one  of  the  forms 

3fi  ±  i,  3fi  ±  i,  3) 

and   f         ,  .]  =  —  i,   when  a  +  bi  is   a  primary^ prime   contained   in   one 

of  the  forms 

SH±  (i-f-O,  3/*±  (i— «).  4) 

Combining  these  with  the  results  obtained  in  the  last  example,  we  see 
that  3  -f-  6i  is  a  quadratic  residue  of  all  primary  primes  contained  simul- 
taneously in  the  forms  i)  and  3),  or  simultaneously  in  the  forms  2) 
and  4),  and  their  associates.  On  the  other  hand  3  +  6*  is  a  quadratic 
non-residue  of  all  primary  primes  contained  simultaneously  in  the  forms 
1)  and  4),  or  simultaneously  in  the  forms  2)  and  3).  These  conditions 
may  in  each  case  be  combined  into  a  single  one  by  Th.  12. 

§  19.     Biquadratic  Residues. 

It  is  impossible  to  leave  the  realm  k(i)  without  a  few  words 
as  to  the  history  of  the  first  treatment  of  these  numbers  from 
the  point  of  view  of  the  theory  of  numbers,  marking  as  it  did 
a  distinct  epoch  in  the  development  of  this  branch  of  mathematics. 

On  the  fifth  of  April,  1825,  Gauss  laid  before  the  Royal  Society 
of  Gottingen  a  paper1  upon  the  subject  of  biquadratic  residues,  a 
brief  report2  of  which  is  given  in  the  "  Gottingische  Gelehrte 
Anzeigen"  for  April  11,  1825. 

He  remarks  in  this  report  that :  "  The  development  of  the  gen- 
eral theory  which  requires  a  most  peculiar  extension  of  the  field 
of  the  higher  arithmetic3  is  reserved  for  future  continuation,  only 
those  investigations  being  taken  up  in  this  first  paper  which  can 
be  completely  carried  through  without  this  extension,"  giving 
thereby  a  foretaste  of  a  step  which  was  to  revolutionize  the  theory 
of  numbers ;  a  step,  however,  the  results  of  which  he  did  not  pub- 
lish until  six  years  later. 

In  this  first  paper  Gauss  defines  a  biquadratic  residue  as  fol- 

^heoria  Residuorum  Biquadraticorum :  Commentatio  Prima.  Works, 
Vol.  2,  p.  65. 

2  Ibid.,  p.  165. 

'Italics  are  the  author's.  See  also  H.  J.  S.  Smith:  Report  on  the 
Theory  of  Numbers,  Arts.  24-36;  Works,  Vol.  I,  pp.  70-86,  and  Bach- 
mann:  Die  Lehre  von  der  Kreisteilung,  Vorlesung  12th.  The  reader  is 
especially  advised  to  consult  Gauss'  reports  on  his  two  papers  and  H.  J.  S. 
Smith's  resume. 


206  THE   REALM    k(i). 

lows :  "  An  integer  a  is  called  a  biquadratic  residue  of  the  integer 
p  when  there  exist  numbers  of  the  form  x*  —  a  which  are  divisible 
by  p,  and  a  biquadratic  non-residue  of  p  when  no  number  of  this 
form  is  divisible  by  p"  or  we  may  say,  as  in  Chap.  Ill,  §  34, 
that  an  integer,  a,  is  a  biquadratic  residue  or  non-residue  of  an 
integer,  p}  according  as  the  congruence 

x* —  a  =  o,  modp, 

has  or  has  not  roots. 

Limiting  the  investigation  now  to  the  case  in  which  p  is  a  posi- 
tive prime  of  the  form  411  +  1  and  a  not  divisible  by  p,  all  other 
cases  being  as  he  says  reducible  to  this  one,  he  separates  all 
integers,  a,  not  divisible  by  p,  into  four  classes,  according  as 

fli(p-D  53  1,  f,  —  1,  or  —  j,  mod  p, 
where  /  is  a  root  of  the  congruence 

f2  -f-  1  s=  o,  mod  p. 
Every  integer  of  a  reduced  residue  system,  mod  p,  satisfies  the 
congruence  x?~x  — 1=0,  mod  p,  1 ) 

which  may  be  written 

(**<p-i>  —  1)  O^-1*  — /)  (x^p-v  +  iX**-^  +  /)  =0,  modp,  2) 
where  /,  —  /  are  the  roots  of  the  congruence 
x2  +  1  ==  o,  mod  p. 

Since  the  congruence  1 )  has  exactly  p  —  1  roots,  each  of  the 
four  congruences  into  which  2)  can  be  resolved  has  exactly 
\{p —  1)  roots  and  the  integers  of  a  reduced  residue  system,  mod 
p,  are  seen  to  fall  into  four  classes,  each  containing  J(/> — 1) 
integers,  according  as  they  satisfy  the  first,  second,  third  or  fourth 
of  these  congruences. 

The  first  class  comprises  those  integers  for  which  the  congru- 
ence 1)  is  solvable;  that  is,  the  biquadratic  residues  of  p  (Chap. 
Ill,  Th.  31)  ;  the  third  comprises  those  integers  which  are  quad- 
ratic but  not  biquadratic  residues  of  p ;  the  second  and  fourth 
classes  are  made  up  of  the  quadratic  non-residues  of  p. 


THE   REALM    k(i).  20? 

We  see,  therefore,  that,  as  Gauss  remarks,  all  biquadratic  resi- 
dues of  p  are  also  quadratic  residues  of  p  and  all  quadratic  non- 
residues  of  p  are  also  biquadratic  non-residues  of  p ;  but  that  not 
all  quadratic  residues  of  p  are  biquadratic  residues  of  p.  Gauss 
now  divides  the  investigation,  as  in  the  case  of  quadratic  residues, 
into  two  parts  according  as  p  or  a  is  supposed  given ;  that  is,  ac- 
cording as  we  are  to  find  what  integers  are  biquadratic  residues 
of  a  given  prime  modulus  and  what  non-residues,  or  of  what 
prime  moduli  a  given  integer  is  a  biquadratic  residue,  and  of 
what  a  non-residue. 

The  first  of  these  is  elementary  in  comparison  with  the  second 
and  easily  solved.  Of  the  second  part  he  treats  three  special 
cases,  a  =  —  I,  a=±2,  but  does  nothing  with  the  general  case. 
These  three  special  cases,  however,  he  fully  discusses,  remarking 
upon  the  exceeding  difficulty  of  the  cases  a=±2. 

In  this  connection  H.  J.  S.  Smith  says  :*  "  The  result  arrived 
at  in  the  case  of  2  is  that,  if  p  be  resolved  into  the  sum  of  an  even 
and  an  uneven  square  (a  resolution  which  is  always  possible  in 
one  and  only  one  way),  so  that  p  =  a2  -f-  b2  (where  we  may  sup- 
pose a  and  b  taken  with  such  signs  that  a=  1,  mod  4,  b  =  af,  mod 
/>),  2  belongs  to  the  first,  second,  third  or  fourth  class  according 
as  \b  is  of  the  form  411,  4»  +  1,  411  +2  or  ^n  -f-  3. 

"The  equation  p  =  a2Jrb2  shows  that  p=(a-\-bi)(a  —  bi), 
or  that  p,  being  the  product  of  two  conjugate  imaginary  factors, 
is  in  a  certain  sense  not  a  prime  number.  Gauss  was  thus  led  to  in- 
troduce as  modulus  instead  of  p  one  of  its  imaginary  factors ;  an 
innovation  which  necessitated  the  construction  of  an  arithmetical 
theory  of  complex  imaginary  numbers  of  the  form  a  -J-  bi." 

In  a  paper2  communicated  to  the  Royal  Society  of  Gottingen, 
April  15,  1831,  a  report3  of  which  is  given  in  the  "  Gottingische 
Anzeigen"  for  April  23,  1831,  Gauss  continues  his  investigations 
in  this  subject,  limiting  himself  still  to  the  case  where  p  is  a  posi- 
tive rational  prime  of  the  form  4^+  1,  a  an  integer  not  divis- 
ible by  p. 

1'Works,  Vol.  I,  p.  71. 

2Th.  Res.  Biq,  Com.  Sec,  Works,  Vol.  II.  §93- 

3  Ibid.,  p.  169. 


208  THE   REALM    k{%). 

He  obtains  by  induction,  but  does  not  prove,  theorems  concern- 
ing the  moduli  of  which  certain  special  values  of  a  (±  3,  5,  — 7, 
— 11,  13,  17,  — 19,  — 23)  are  biquadratic  residues,  and  those  of 
which  they  are  non-residues,  but  says  in  the  above  mentioned 
report:  "Although  all  these  special  theorems  can  be  discovered 
so  easily  by  induction  it  appears  nevertheless  extremely  difficult 
to  find  a  general  law  for  these  forms,  even  if  much  that  is 
common  makes  itself  evident,  and  it  is  still  more  difficult  to  find 
proofs  for  these  theorems.  The  methods  used  for  the  num- 
bers 2  and  — 2  in  the  first  paper  can  not  be  applied  here,  and 
if  other  similar  methods  such  as  that  used  in  dealing  with  the 
first  and  third  classes,  could  serve  to  solve  the  problem,  they 
prove  themselves,  however,  entirely  unsuitable  as  foundations 
for  complete  proofs.  One  soon  recognizes,  therefore,  that  it  is 
only  by  entirely  new  paths  that  one  can  penetrate  into*  this  rich 
domain  of  the  higher  arithmetic.  The  author  has  already  pointed 
out  in  the  first  paper  that  for  this  purpose  a  peculiar  extension  of 
the  field  of  the  higher  arithmetic  is  indispensable,  without,  how- 
ever, explaining  more  fully  wherein  this  consisted ;  the  design 
of  the  present  paper  is  to  make  known  the  nature  of  this  extension. 
It  is  simply  that  a  true  basis  for  the  theory  of  the  biquadratic 
residues  is  to  be  found  only  by  making  the  field  of  the  higher 
arithmetic,  which  usually  covers  only  the  real  whole  numbers, 
include  also  the  imaginary  ones,  the  latter  being  given  full  equal- 
ity of  citizenship  with  the  former.  As  soon  as  one  has  per- 
ceived the  bearing  of  this  principle,  the  theory  appears  in  an 
entirely  new  light,  and  its  results  become  surprisingly  simple." 

This  widening  of  the  field  of  the  higher  arithmetic  consists, 
then,  in  considering  our  integers  to  be  all  those  numbers  of  the 
form  a  +  bi,  in  which  a  and  b  are  rational  integers.  The  defini- 
tions of  divisibility,  prime  number,  etc.,  and  the  principal  theo- 
rems relating  to  rational  integers  having  been  shown  to  have  their 
analogues  for  the  integers  of  this  extended  system,  our  realm 
k(i),  as  has  been  proved  in  the  preceding  pages,  Gauss  then 
develops  briefly  the  theory  of  quadratic  residues  for  the  integers 
of  this  new  system.     Passing  to  the  subject  of  biquadratic  resi- 


THE   REALM    k(i).  20O, 

dues,  he  separates  all  integers  not  divisible  by  a  given  modulus 
into  four  classes,  as  follows  : 

"  If  the  modulus  be  a  complex  prime  number,  a  +  bi,  where  a 
is  always  assumed  odd,  b  even,  and  k  a  complex  number  not 
divisible  by  a  +  bi,  then,  for  the  sake  of  brevity  p  being  written 
for  a2  -+-  b2,  we  have  in  all  cases 

£i(p-D  ==  i}  i}  —  i}  —  i}  mod  a  -\-  bi, 

and  thereby  all  numbers  not  divisible  by  a  4-  bi  are  separated  into 
four  classes,  to  which  in  the  above  order  the  biquadratic  charac- 
ters 0,1,2,3  are  ascribed."  That  is,  the  biquadratic  character 
of  an  integer,  k,  with  respect  to  a  prime  modulus,  a  -f-  bi,  is  the 
exponent  of  the  lowest  power  of  i  to  which  ki{p~x)  is  congruent, 
mod  a  +  bi,  where  p  =  a2  +  fr2. 

"  It  will  be  observed  that,  when  a  +  bi  is  a  prime  of  the  first 
degree,  the  fourfold  classification  of  the  real  residues  of  a  +  bi 
which  we  thus  obtain  is  identical  with  that  obtained  for 
p,  =n[a-\-bi],  in  the  real  theory;  for  the  numbers  /  and  — f, 
being  the  roots  of  the  congruence 

x2  -f-  1  =  o,  mod  p, 

satisfy  the  same  congruence  for  either  of  the  complex  factors, 
a-\-bi,  a  —  bi,  of  p,  and  are  therefore  congruent  respectively  to 
+  i  and  — i,  for  one  of  these  factors,  and  to  — i  and  -f  i  for  the 
other.1 

"  Evidently  the  character  o  belongs  to  the  biquadratic  residues, 
the  remaining  ones,  1,  2,  3,  to  the  biquadratic  non-residues,  to  the 
character  2  corresponding  quadratic  residues,  to  the  characters 
1  and  3  on  the  other  hand  quadratic  non-residues. 

"  One  recognizes  at  once  that  it  is  only  necessary  to  determine 
this  character  for  such  values  of  k  as  are  themselves  complex 
primes,  and  here  induction  leads  immediately  to  most  simple  re- 
sults. If,  first  of  all,  we  put  k  =  1  +  i,  it  is  seen  that  the  charac- 
ter of  this  number  is  always  congruent  to 

i( — a2-f  2ab  —  3&2-f-i),  mod 4, 

1  See  H.  J.  S.  Smith :  Works,  p.  197. 
14 


2IO  THE   REALM    k(i). 

and  similar  expressions  are  found  for  the  cases  k  =  i  —  i,  —  I  -\-.% 
—  i  —  i. 

"  If,  on  the  other  hand,  k  be  such  a  prime  number  c  +  di,  that 
c  is  odd  and  d  even,  we  can  obtain  by  induction  a  reciprocity  law 
quite  analogous  to  the  fundamental  theorem  for  quadratic  resi- 
dues ;  this  theorem  can  be  expressed  most  simply  in  the  following 
manner : 

"  If  c  -\-  d  —  i  as  well  as  a-\-b  —  i  be  divisible  by  4  (to  which 
case  all  others  can  be  easily  reduced),  and  the  character  of  the 
number  c  -f-  di  zinth  respect  to  the  modulus  a  -f-  bi  be  denoted  by 
llf  that  on  the  other  hand  of  a-\-  bi  with  respect  to  the  modulus 
c  -f-  di  by  l2,  then  h  —  l2  when  one  (or  both)  of  the  numbers  d 
and  b  is  divisible  by  4;  on  the  other  hand  lx  —  l2  ±  2,  when  neither 
of  the  numbers  d,  b  is  divisible  by  4. 

"  These  theorems  contain  in  truth  all  the  essentials  of  the  theory 
of  the  biquadratic  residues ;  easy  as  it  is  to  discover  them  by 
induction,  it  is  most  difficult  to  prove  them  rigorously,  especially 
the  second,  the  fundamental  theorem  of  the  biquadratic  residues. 
On  account  of  the  great  length  of  the  present  paper  the  author 
finds  himself  obliged  to  postpone  to  a  third  paper1  the  presenta- 
tion of  a  proof  of  the  latter  theorem,  which  has  been  in  his  pos- 
session for  twenty  years.  On  the  other  hand,  the  present  paper 
contains  the  complete  proof  of  the  first  theorem  relating  to  the 
number  1  +  h  upon  which  are  dependent  the  theorems  relating  to 
1  —  i,  —  I  -(-  i,  —  1  —  i.  This  proof  will  give  some  idea  of  the 
complexity  of  the  subject." 

The  above  will  be  made  plainer  to  the  reader  by  the  following 
brief  resume.     The  integer  a  is  said  to  be  a  biquadratic  residue 

1  Gauss  never  published  his  proof  of  this  theorem,  but  soon  after  the 
theorem  was  published  Jacobi  succeeded  in  proving  it,  and  communicated 
this  proof  to  his  pupils  in  his  lectures  at  Konigsberg  in  the  winter  of 
1836-37.  He  did  not,  however,  publish  his  proof,  and  the  first  published 
proofs  are  by  Eisenstein,  who  gave  in  all  five.  See  Crelle,  Vol.  XXVIII, 
P-  53,  P-  223,  and  Vol.  XXX,  p.  185;  also  H.  J.  S.  Smith:  Works,  Vol. 
I,  p.  78,  and  Bachmann:  Die  Lehre  von  der  Kreisteilung,  p.  168. 


THE   REALM    k(i). 


211 


or  non-residue  of  a  prime,  *,  a  being  prime  to  tt,  according  as  the 

congruence  x*  =  a,  mod7r, 

is  or  is  not  solvable. 
From  Th.  16  we  have 

a  w[7r]-1  =  if  modTT,  3) 

and  since,  excluding  the  case  ir=  r  +  h1  n[7r]  —  I  is  always  divis- 
ible by  4,  we  may  write  3)  in  the  form 

/    nQ]-l  \     /     n  [*•]-!  \     /     n[ir-]-l  \    /    n[n^-l  \ 

[a     4      -i)\a     4     -i)\a     4      + 1)  \a     4     +*j=o,  modTr, 
each  of  the  congruences 

»[tt]-1 


t        4 

"L*]-i 
t      4 

w[tt3-1 
4 


,  mod7r, 


a     -       ss  —  % 
which  may  be  written  in  the  common  form 

n\_n]-l 


a    4      wmif,  modTr,  r  =  o,  1,2,  3, 

is  seen  to  have  exactly  (n[Tr]  — 1)/4  incongruent  roots,  and  the 
integers  of  a  reduced  residue  system,  mod  ir,  fall  into  four  classes 
according  as  they  satisfy  the  first,  second,  third  or  fourth  of 
these  congruences. 

The  integers  of  the  first  class  are  the  biquadratic  residues  of  it, 

for  a     4     =  1,  modTr, 

is  the  necessary  and  sufficient  condition  that  a  shall  be  a  biquad- 
ratic residue  of  7r. 

The  integers  of  the  first  and  third  classes  are  together  the 
quadratic  residues  of  tt,  for  they  are  the  roots  of  the  congruence 

a     2     =1,  modTr. 

1  It  is  easily  seen  that  every  integer  not  divisible  by  1  -f-  *  is  a  biquadratic 
residue  of  1  +  i. 


212  THE   REALM    k(J). 

The  integers  of  the  second  and  fourth  classes  are  together  the 
quadratic  non-residues  of  if,  for  they  are  the  roots  of  the  congruence 

n[tr]-l 

a     2     =  —  i ,  mod  7T. 
The  exponent  of  the  power  of  i  for  which  the  congruence 

n[ir]-l 

a     4     ■■  ir,  mod  r,  r  =  o,  i ,  2,  3 
is  satisfied  is  called  the  biquadratic  character  of  a  with  respect 
to  it  and  this  power  of  i  is  denoted  by  the  symbol  (o0r)4,  so  that 
we  have  always  % 


©.«~ 


mod 


The  symbol  (a/7r)4,  which  is  due  to  H.  J.  S.  Smith,  seems  preferable 
to  ((ct/7r)),  which  was  adopted  by  Jacobi,  as  by  a  change  of  subscript 
it  will  serve  for  the  theory  of  residues  of  other  degrees. 

If  now  (a/ir)  have  the  meaning  previously  assigned,  we  see 
easily  that 


a)-©;- 


If  we  understand  by  the  quadratic  character  of  a,  mod  *",  instead  of 
1  or  —  1,  the  exponent  of  the  lowest  power  of  —  1  to  which  a  is  con- 
gruent, mod  t,  the  notation  for  quadratic  residues  will  be  brought  into 
accordance  with  that  given  above  for  biquadratic  residues. 

The  symbol  (fit/*) 4  obeys  the  following  laws: 
From  ax  =  a2,  mod  r,  it  follows  that 


©.-(?).- 


If  ax  and  a2  be  two  integers,  which  may  be  equal,  not  divisible 
by  w,  then  from 

and  (  — )    =a2    4     f  mod  7r, 

(5),(5),-<°.«.> 


it  follows  that 


nOJ-l 

4    ,  mod  7T, 


THE   REALM    k(i).  21  3 

(¥).-  (5),  (5);, 

Since  every  integer  a  can  be  written  in  the  form 

a  =  ir(i+i)*Plp2--Pn, 

where  r  =  o,  1,2,  3;  ^  =  0,  or  a  positive  integer;  and  p1,p2,---,pn 
are  odd  primary  primes,  we  have 

W4  =  W4  VflT  j4  Vtt j4  w4 * '  •  w; 

and  the  determination  of  the  value  of  [  -  |    is  seen  to  be  resolved 

Wi 

into  the  determination  of  the  values  of  ( -  |  ,  I |   and  1  —  1 

where  p  is  an  odd  primary  prime. 

The  following  theorem  gives  a  simple  criterion  for  determining 
the  value  of  (*/r)«: 

Theorem  21.  If  Tr  =  a-\-bi  be  an  odd  primary  prime,  then  i 
has  the  biquadratic  character  o,  1,  2  or  3  with  respect  to  the  mod- 
ulus tt,  according  as  we  have  a^=i,  7,  5  or  3,  mod  8;  that  is, 


\a  +  bi) 


Since  a  +  bi  is  an  odd  primary  prime,  we  have  either 
a  =  4k  +  1 ;  b  =  4k, 
or  a  =  4k  +  $',  b  =  4k+2, 

and  hence 


\a  -f  bi) 


q2+&2_l 

i     4      =/2fe,  when  0  =  4^+1, 


=  j2&+3,  when  a  =  4^-(-3. 
But  2^  =  oor  2,  mod  4,  according  as  k  is  even  or  odd ;  that  is, 
according  as  0  =  4^+1  =  1  or  5,  mod  8 ; 

and  2k  +  3  =  3  or  1,  mod  4,  according  as  k  is  even  or  odd ;  that 
is,  according  as 


214  THE  REALM  k{i). 

a  =  4k  +  3  =  3  or  7,  mod  8. 
Hence  ( — ^-.)  ==1,*,—  ior  —  i. 

according  as  a=  1,  7,  5,  or  3,  mod  8. 

The  following  table  gives  the  biquadratic  character  of  i  with 
respect  to  each  odd  primary  prime  whose  norm  is  less  than  50. 


Biq.   Char. 

Odd  Primary  Primes. 

O 

% 

i+4*,  i—4*,  —7- 

I 

—  I  +  2f,  —  I  —  21,  —  1+6*',  —  I  - 

-6*. 

2 

—  3,5  +  4h  5—  4*. 

3 

3  +  2»,  3  —  21,  —  5  +  2f,  —  5  —  2*. 

The  following  theorem  gives  the  biquadratic  character  of  1  +  * 
with  respect  to  an  odd  primary  prime  modulus. 
Theorem  22.     If  a-\-  bi  be  any  odd  primary  prime 


(i±l\  _;~bi*-1 

\a  +  bi)-1 


For  the  proof  of  this  theorem  see  Gauss:  Works,  Vol.  II,  p. 
135;  Eisenstein:  Crelle,  Vols.  28  and  30;  Bachmann:  Die  Lehre 
von  der  Kreisteilung,  p.  181. 

The  following  table  gives  the  biquadratic  character  of  1  +  f 
with  respect  to  each  odd  primary  prime  whose  norm  is  less  than  50. 


Biq.  Char. 

Odd  Primary  Primes. 

O 

3  —  2i,5  +  4i  —  1—  61 

I 

1  —  4h  —  5  +  2i  —  1  +  6*  • 

2 

—  1 +2*,  —  5  —  2f,5  —  4*,  —  7. 

3 

—  1  —  2%,  —  3, 3  +  2%,  1  +  4*. 

This  theorem  is  easily  seen  to  be  the  equivalent  of  Gauss'  (p. 
209),  for  although  the  modulus  is  here  restricted  to  an  odd  primary 


THE   REALM    k(i).  21  5 

prime,  a-\-bi,  while  in  Gauss'  it  can  be  either  ±{a-\-bi),  where 
a  -f-  bi  is  an  odd  primary  prime,  this  makes  no  difference,  since 


(i+i)  _  (■**' ) 


We  have  only  to  show  therefore  that 

\{—  a*  +  2ab  —  3b*+i)^i(a—  b  —  b2—  i),  mod4,      4) 

where  a  =  1,  b  =  o,  or  a  ==  —  1,6  =  2,  mod  4. 

Putting 

a  =  4a1-\-i,b  =  4b1,  or  0  =  4^  —  i,fr  =  4&1  +  2 

in  4),  we  obtain  in  both  cases 

(b1  —  a1)(2a1-{-2b1+i)=a1  —  b1,  mod4; 

that  is  (»i  +  ^i  +  I)(fli  —  &i)=o,  mod 2, 

is  a  necessary  and  sufficient  condition  that  4)  shall  hold,  and  this 
condition  is  easily  seen  to  be  satisfied  by  all  values  of  ax  and  bv 

The  value  of  (a/7r)4  is  determined  by  means  of  the  reciprocity 
law  given  by  Gauss,  which  can  be  expressed  most  simply  as 
follows : 

Theorem  23.  The  biquadratic  characters  of  two  odd  primary 
primes  of  k(i)  with  respect  to  each  other  are  the  same  or  opposite 
according  as  one  of  the  primes  is  =1,  mod  4,  or  both  are 
=  j  -\-  2i}  mod  4. 

This  can  be  expressed  symbolically  as  follows : 

£).-<->---m 

in  which  ir  and  p  are  any  two  odd  primary  primes  of  k(i). 


Since 
we  have  from  the  last  theorem 


(>)-(;):• 


(>)-(& 


and  from  this  can  easily  deduce  the  quadratic  reciprocity  law  as 
given  in  Th.  20. 


2l6  THE   REALM    k  (i)  . 

The  biquadratic  character  o  is  opposite  to  2,  and  I  to  3,  this 
corresponding  to  lt  =  l2  ±  2  in  Gauss'  theorem  (p.  210) .  His  con- 
dition, that  a  +  b  —  1  and  c  +  d  —  1  shall  both  be  divisible  by  4, 
is  evidently  satisfied  when  the  primes  are  primary.  Furthermore, 
it  is  easily  seen  from  the  definition  (p.  194)  that  every  odd  pri- 
mary prime  is  5=  1  or  3  +  21,  mod  4 ;  and  this  is  equivalent  to 
Gauss'  condition  that  b  (or  d)  be  divisible  or  not  divisible  by  4. 

Ex.  1.     To  determine  the  value  of 

\5  +  4*A 

Resolving  1  +  31  into  its  primary  prime  factors,  we  have 

\S  +  4*A       \S  +  4*/A5  +  4«/A    5  +  4*  )i 

ByTh.  21  (-i—\   =i3{b-1)/2  =  ?, 

\5  +  4*/4 

and  by  Th.  22  (±+1  \  =  i°. 

\5  +  4*7 

Since  — 1+2*  and  5  -f-  41  are  odd  primary  primes  we  have  by  Th.  23 

and  since  5  +  4*  —  (1  —  3*)  (—  1  +  2i)=  —  i;1 

that  is,  5  -f  41  ==  —  i,  mod  —  1  -f  2/, 

we  have  (_*  +  4L  \        /     -»'      \        / 1_Y  =  * 

V—  I  4- 2*  A         \— I+2I/4         V—  I+2t/4 

Combining  these  results,  we  have 

\5  +  4*A 

that  is,  1  +  31  is  a  biquadratic  non-residue  of  5  -f  41,  or  in  other  words  the 
congruence  x*  ==  1  -f-  3/,  mod  5  -f  41 

has  no  roots. 

We  see  also  that  1  +  3*  is  a  quadratic  non-residue  of  5  -f-  41. 

Ex.  2.  To  classify  the  odd  primary  primes  of  k(i)  according  to  the 
biquadratic  character  of  —  1  -f  2f  with  respect  to  each  of  them. 

Let  it  be  any  odd  primary  prime  of  k(i). 

1  We  select  1  —  31*  as  a*  is  chosen  in  Th.  A. 


THE   REALM    k(i).  217 

We  have  two  cases  to  consider  according  as  t^i  or  3  +  21,  mod 4. 

i.  ''"^i,  mod  4. 

Then 

V       •        /*~~  \—  1  +  2i74—  V—  1  4-  2iji     \-i+  21) \  '  \—  1  +  2i) t 
or(      -*     )    , 

V— 1  +  21/4 

according  as  ■'as  I.  i  —  1  or  —  i,  mod  —  1  +  21,  1,  *',  —  1,  —  t  being  a 

reduced  residue  system,  mod  —  1  +  **> 

But 

fcfe)  =«■  fcira).-*  G^a)-'-  (^),=<'3- 

Hence  with  respect  to  an  odd  primary  prime,  wt  sp  i,  mod  4,  —  1  +  2t  has 
the  biquadratic  character  o,  1,  2  or  3,  according  as  we  have  *"==i,  t,  — 1, 
or  —  i,  mod  —  1  +  21. 

ii.  *  35  3  +  2»,  mod  4. 

Since  we  have  both  »  and  —  i+2i  =  3-f  2#,  mod  4,  it   follows  that 


(-*+*) (    '   Y 

V       *       A  V— 1+21/4 


Hence  with  respect  to  an  odd  primary  prime,  t,  ss3  -f 2'j  mod  4, 
—  1  +  21  has  the  biquadratic  character  o,  I,  2  or  3,  according  as  we 
have  *  as  —  1,  —  *j  1  or  i,  mod  —  1  +  2/. 

Combining  these  conditions  we  see  that  — 1+2*  has  with  respect  to 
an  odd  primary  prime,  *",  the  biquadratic  character 

0  where  ir=n( —  4  +  &')  +  1  or  A*( —  4  +  80  +  3  +  21, 

1  where  t  —  fi( — 4  -|-  &")  +  1  -j-  4*  or  /*( — 4  +  Si)  +  3"  —  2», 

2  where  7r  =  /i(— 4-f  81)  +  1— 4/  or  /*(— 4  + &*) +3 +  6*, 

3  where  7T  — /u( — 4  +  8O — 3  or  /*( — 4  +  8*)  +7  +  21, 

M  being  any  integer  of  k{i). 

Ex.  3.     Determine  whether  the  congruence 

x*  =  9  +  7i,  mod  5 +  4^ 
has  roots. 

Ex.  4.     Class   the   odd   primary   primes   of    k(i)    according   to    the   bi- 
quadratic character  of  3  +  i  with  respect  to  each  of  them. 


CHAPTER  VI. 

The  Realm  &(V — 3)- 

§  i.    Numbers  of  &(V — 3). 

The  number  V — 3  is  defined  by  the  equation 

s2+3==0,  I) 

which  it  satisfies.  We  can  show  exactly  as  in  k(i)  that  all  num- 
bers of  &(V — 3)  have  the  form  a  +  &V — 3>  where  a  and  b  are 
rational  numbers.  The  other  root,  — V — 3>  of  1)  defines  the 
realm  k( — V — 3)  conjugate  to  &(V — 3).  These  two  realms 
are,  however,  evidently  identical.  The  number  a',  —  a — b\/ — 3, 
obtained  by  putting  — V — 3  for  V  —  3  in  any  number  a, 
=  a-\-by/ — 3,  of  &(V — 3),  is  the  conjugate  of  a;  for  example, 
2+V — 3  and  2 — V — 3  are  conjugate  numbers. 
y^>  A  rational  number  considered  as  a  number  of  &(V — 3)  is  evi- 
^jA  dently  its  own  conjugate.  The  product  of  any  number,  a,  of 
&(V — 3)  by  its  conjugate  is  called  its  norm,  and  is  denoted  by 
n[a]  ;  that  is, 

n[a  +  b  V=z\  =  (a  +  b  y- ^)(fl  —  &V=3)=a2+3&2 
We  see  that  the  norms  of  all  numbers  of  k{  V — 3)  are  positive 
rational  numbers.     We  can  prove  exactly  as  in  k(i)   that  the 
norm  of  a  product  is  equal  to  the  product  of  the  norms  of  its 
factors;  that  is, 

n[ap]=n[a]n[/3], 

where  a  and  ft  are  any  numbers  of  &V — 3- 

We  observe,  just  as  in  k(i),  that  every  number  a,  ==a  -f-  b\/ — 3, 
of  &(V — 3)  satisfies  a  rational  equation  of  the  second  degree, 
that  being  the  degree  of  the  realm,  and  that  this  equation  has  for 
its  remaining  root  the  conjugate  of  a. 

The  numbers  of  &(V — 3)  fall  then,  as  in  k(i),  into  two  classes, 
imprimitive  and  primitive,  according  as  the  above  equation  is 

218 


THE    REALM    k(\/ 3).  219 

reducible  or  irreducible;  that  is,  according  as  6  =  or=j=o.     The 

imprimitive  numbers  are  therefore  the  rational  numbers,  and  the 

primitive  numbers  all  the  other  numbers  of  the  realm. 

It  is  evident  that  any  primitive  number  of  &(V — 3)  can  be 

taken  to  define  the  realm. 

This  realm  as  well  as  the  following  ones  will  not  be  discussed  as  fully 
as  k(i).  Our  desire  is  merely  to  bring  out  those  points  of  difference 
from  k(i)  which  necessitate  some  change  in  our  conceptions,  and  to 
show  that  after  these  changes  have  been  made  and  the  unique  factoriza- 
tion theorem  proved  for  the  integers  of  the  realm,  we  can  get  as  in  k(i) 
a  series  of  theorems  analogous  to  those  for  rational  integers. 

§  2.    Integers  of  k  ( V — 3). 

To  determine  what  numbers  of  £(V — 3),  in  addition  to  the 
rational  integers,  are  algebraic  integers,  we  observe  that  as  in  k(i) 
the  necessary  and  sufficient  conditions  that  any  number,  a, 
=  a-\-b^ — 3,  of  &(V — 3)  shall  be  an  integer  are 

a  +  a!  =  a  rational  integer, 
and  aa'  =  3.  rational  integer. 

If  we  write  a  in  the  form 

ai  +  bi  ^ -1 

where  a  =  a1/c1,  and  b  =  b1/c1,  ax,  blt  ct  being  integers  with  no 
common  factor,  these  conditions  become 

«t  +  ^i  V  — 3      ^-^iV-3       2a.  .         ■  . 

-1        * +  -3 ! =  — '  =  a  rational  integer,     1) 

cx  cx  v 

1      l   )  (  )  =2      =a  rational  integer.     2) 

One  at  least  of  the  three  following  cases  must  occur: 
i.  c1=\=2or  1;       ii.  cx  =  2\       iii.  c1  =  i. 

i.  The  impossibility  of  i  is  proved  as  in  k{i). 

ii.  If  q  =  2,  2a1/c1  can  be  an  integer,  and  yet  ax  not  contain  the 
factor  2,  ax2  -f-  2>bx2  being  divisible  by  22  when  ax  and  &2  are 
both  odd. 


220  THE   REALM    &(V 3). 

Hence  c1  =  2,  in  which  case  a±  and  bx  must  both  be  odd; 
or   ^=1.     Hence   every   integer   of    &(V — 3)    has   the    form 

J(°  +  ^V — 3) >  where  a  and  b  are  either  both  odd  or  both  even, 
and  all  numbers  of  this  form  are  integers. 

§3.     Basis  of  MV^).1 

A  basis  of  k(  V — 3)  w  defined  as  in  k(i).  It  will  be  observed 
that  the  integer  V —  3  defining  k  ( V —  3 )  does  not  constitute  with 
1  a  basis  of  the  realm  as  i  and  1  did  in  k(i)  ;  that  is,  there  are 
integers  of  the  realm  that  can  not  be  represented  in  the  form 
x  -f-  y  V —  3,  where  x  and  37  are  rational  integers.  We  shall  see, 
however,  that  two  integers  of  &(V — 3)  can  be  found,  which 
form  a  basis  of  the  realm.  For  example,  1,  £( —  1  +  V —  3)  1S  a 
basis  of  k{  V — 3)  ;  for  let  J( —  1  +V — 3)>  which  is  seen  to  be  an 
integer,  be  represented  by  p,  and  -J (a  +  ^V — 3)  be  any  integer  of 
k  (  V —  3  ) .  We  shall  show  that  J  ( a  +  b  V —  3  )  can  be  put  in  the 
form  x  +  yp,  where  x  and  y  are  rational  integers. 

_,  a  -f  b  v  —  3  2x—y      y     

put       — - — 5-  =  *+j,P — -Tz+iv_3> 

which  gives  2x  —  y  =  a,  y  =  b, 

whence  x  =  ^(a-\-  b),  y  =  b, 


a  ,u      £  a  +  bv  —  3      a  +  b 

and  therefore  ~  = 1-  bo 

2  2  r' 

where  \{a-\-b)  is  a  rational  integer,  since  a  and  b  are  either  both 
even  or  both  odd.  Every  integer  of  &(V — 3)  can  be  repre- 
sented therefore  in  the  form  x  -\-  yp,  where  x  and  y  are  rational 
integers;  that  is,  1,  p  is  a  basis  of  &(V — 3).  Moreover,  every 
number  of  the  form  x  -f-  yp  can  be  put  in  the  form  J(a  +  by/ — 3), 
where  a  and  b  are  both  odd  or  both  even,  and  hence  is  an  integer 
of  &(V — 3).  For,  supposing  x  and  3;  known,  and  a  and  b  un- 
known, we  see  from  the  above  analysis  that  a  and  b  will  be  either 
both  odd  or  both  even,  according  as  y  is  odd  or  even.  The  sum, 
difference  and  product  of  any  two  integers  of  &(V — 3)  is  an 
integer  of  &(V — 3),  for 

1  See  Chap.  V,  §  3. 


THE  REALM    k(\/ 3).  221 

O  +  yP)  =b  (*,  +  ylP)  =  x±x1+(y±  yt)p, 
and 

O  +  yp)  (*i  +  y\p)  —x*t  +  (*y,  +  *i:y)p  +  yjip2 

=  **x  —  3'3'i  +  O^i  +  xxy  —  vyjp, 

since  p2  -f  p  -f-  i  =  o. 

§4.    Conjugate  and  Norm  of  an  Integer  of  &(V— ^3)- 

The    conjugate    of    p    is    p'=|( — i  —  V — 3)  =p2.       Since 

P  +  p'  =  P  +  p2  =  — 1,  and  pp'  =  p3=i,  p  satisfies  the  equation 

^r2  +  .r+i=o; 
that  is,  p  and  p2  are  the  imaginary  cube  roots  of  unity ;  therefore 
fc(V — 3)  is  called  the  realm  of  the  cube  roots  of  unity.  If 
a,  =a-{-bp,  be  any  integer  of  &(V — 3)»  its  conjugate  is  a', 
=  a-\-bp2.  The  conjugate  of  a -\- bp2  is  evidently  a-\-bp*, 
=  a  +  bP. 
Hence  n[a]  =  (a-\-  bp)  (a  +  bp2) 

=  a*  +  ab(p  +  P*)+by 

=  a2  —  ab  +  b2, 
which  is  seen  to  be  a  positive  integer. 
For  example 

«[3  +  2p]=9  — 6  +  4  =  7- 
§5.    Discriminant  of  £(V — 3).1 
The  discriminant  of  k{  V — 3)  is  the  squared  determinant 

1     P 
I     P2 

formed  from  a  pair  of  basis  numbers  and  their  conjugates. 
Denoting  it  by  d,  we  have 

d  =  -3- 

§6.    Divisibility  of  Integers  of  fc(V — 3)- 

We  define  the  divisibility  of  integers  of  fc(V — 3)  exactly  as 
we  defined  that  of  the  integers  of  R  and  &(/),  and  all  that  fol- 
lowed from  this  definition  in  R  and  k(i)  holds  for  k(\/ — 3). 

1  See  Chap.  V,  §§3,  4 ;  the  same  remarks  hold  here. 


222  THE   REALM    &(\/ 3). 

Ex.  i.    We  see  that  4  +  5P  is  divisible  by  3  +  2p,  since 

4+5P=  (3  +  2p)(2  +  p) 

=  6  +  79  +  2P2 

=  4  +  SP, 
since  p2  =  —  1  —  p. 

Ex.  2.  On  the  other  hand,  5  -f-  2/>  is  not  divisible  by  3  +  p,  since  there 
exists  no  integer  of  k(y/  —  3)  which  when  multiplied  by  3  +  P  gives 
5  +  2p;  for  let 

$  +  2p=  (3  +  p)(*  +  yp) 

=  3*+  (*  +  3y)p  +  yp2  1) 

=  3*  — y  +  0  +  2y)p; 
thus  x  and  3/  must  satisfy  the  equations 

$x  —  y  =  Sj  x-\r2y  =  2, 
which  give  x=  12/7,  y  =  y7;  that  is,  1)  does  not  hold  for  integral 
values  of  x  and  y,  and  hence  5  +  2p  is  not  divisible  by  3  +  p. 

Theorem  i.  If  a  be  divisible  by  (3,  then  n[a]  is  divisible  by 
n[jB].  0 

For  from  a  =  fiy  follows  n[a]=n[/3]n[y]  ;  that  is,  n[a]  is 
divisible  by  n[/3].  As  was  seen  in  k(t),  the  converse  of  this 
theorem  is  not  in  general  true. 

A  common  divisor  of  two  or  more  integers  is  defined  as  in 
R  and  k(i). 


§  7.    Units  of  k ( V  —  3) .    Associated  Integers. 

The  units  of  &(V — 3)  are  defined,  as  in  the  case  of  the  last 
two  realms,  as  those  integers  of  &(y — 3)  that  divide  every 
integer  of  the  realm.  They  therefore  divide  1,  and  since  every 
divisor  of  1  is  evidently  a  unit,  the  units  may  also  be  defined 
either  as  those  integers  of  &(V — 3)  whose  reciprocals  are  also 
integers  of  k{  V — 3),  or,  since  if  c  be  a  unit,  n[e]  must  divide  1, 
as  those  integers  of  k(y — 3)  whose  norms  are  1. 

To  determine  the  units  of  k(  V — 3)  we  let  «,  =•*■  +  37>,  be  one 
of  them,  and  put 

n[e]=x2  —  xy  +  y2=(x  —  iy)*  +  iy*  =  i, 

from  which  we  see  that  y  can  have  only  the  values  o,  1  and  —  1. 


THE   REALM    k(\/ 3).  223 

y=      o  gives  x2=i,  x=i  or  — I,  and  hence 

€=:  1  or  —  1 ; 

3/=      1  gives  x2 —  jr.  4- 1  as  i,     .r  =  o  or        I,  and  hence 

€  =  p,  or  1  +p  =  —  p2; 

y== — 1  gives  x2  +  x  +  I  =  I,     ,r  =  o  or  — i,  and  hence 

€  =  —  p,  or  — 1 — p  =  p2. 

Hence  €  can  have  any  one  of  the  six  values  ±  i,  ±  p,  ±  p2,  which 
are  therefore  the  units  of  fc(V — 3)- 

As  &(V — 3)  contains  the  primitive  sixth  roots,  i(i  +  V — 3)  an<3 
i(i  —  V  — 3),  of  1,  and  hence  the  cube  roots  of  1,  it  might  more  properly 
be  called  the  "  realm  of  the  sixth  roots  of  unity."  Taking  1,  «, 
=  l(i-f-V — 3),  as  a  basis,  we  would  have  as  the  six  units  of  the  realm 
1,  w}  «2,  <a3  —  —  1,  w4,  w°,  the  six  sixth  roots  of  unity. 

The  nomenclature  used  above  is,  however,  the  usual  one,  and  hence 
has  been  adopted  here. 

If  two  integers,  a  and  /?,  have  no  common  divisor  except  the 
units,  they  are  said  to  be  prime  to  each  other,  or,  excluding  the 
units,  to  have  no  common  divisor. 

The  six  integers,  a,  — a,  pa,  — pa,  p2a,  — p2a,  obtained  by  mul- 
tiplying any  integer,  a,  of  &(V — 3)  by  the  six  units  in  turn,  are 
called  associated  integers;  for  example,  the  six  integers,  1  —  6p, 

—  1  -f-  6p,  6  +  7p,  —  6  —  jp,  —  7  —  p  and  7  +  p  are  associated. 
Any  integer  which  is  divisible  by  a  is  also  divisible  by  — a,  pa, 

—  pa,  p2a  and  — p2a.  Hence  in  all  questions  of  divisibility,  asso- 
ciated integers  are  considered  as  identical;  that  is,  two  factors, 
one  of  which  can  be  changed  into  the  other  by  multiplication  by 
a  unit,  are  looked  upon  as  the  same. 

§8.    Prime  Numbers  of  &(V  — 3). 

The  definitions  are  identical  with  those  in  k(i). 

We  can  determine  whether  any  integer  of  &(V — 3)  is  prime 

or  composite  by  the  method  employed  for  the  same  problem  in 

k(i),  the  process  depending  upon  Th.  1. 

Ex.   1.     To  determine  whether  2  is  a  prime  or  composite  number  of 
kW-3). 
Put  2=(a  +  bp)(c  +  dp); 

then  4=  (a2  —  ab  +  b2)  (c2  —  cd  +  <P) , 


+^=2, 


224  THE    REALM    &(V 3). 

whence  we  have  either 

a2  —  ab  +  b2z=2,  c2  —  cd -\- d2  =  2,  1) 

or  a2 —  ab-\-b2=i,c2  —  cd-{-d2  =  4.  2) 

It  is  easily  seen  that  1)  is  impossible;  for,  if 

then  I  &  I  =  1      and  similarly      [  a  |  $x.  3) 

It  is  evident  that  no  pair  of  values  of  a  and  b,  which  fulfil  the  condition 
3),  can  satisfy  1).  Hence  1)  is  impossible,  and  2)  is  the  only  admissible 
case ;  that  is,  a  +  bp  is  a  unit.  Therefore  2  is  a  prime  number  in 
fc(V~"3). 

Ex.  2.     To  determine  whether  3  is  a  prime  or  composite  number  of 

KV-T). 

Put  3  =  (a  +  &p)  (c  +  dp)  ; 

then  9—(a2  —  ab  +  b2)  {c2  —  cd  +  cP). 

whence  we  have  either 

a2  —  ab  +  b2  =  3,  <r  —  cd  +  d2=%  4) 

or  a2  —  ab-\-b2=  1,  c2  —  cd  +  d2  =  g.  5) 

Now,  if  a2  —  a&  +  6*=i,  a  +  ftp  is  a  unit  and  hence  5)  is  not  an  actual 
factorization. 


If  a2  —  ab  +  b2=(a—  |Y 


4        J 
then  I  &  I  g  2,      and       I  a  I  i  2.  6) 

The  possible  values  of  fc  which  satisfy  6)   are  0,  ±  I,  ±  2.     Considering 
them  in  turn  we  see  that 

b  =  o,  gives  a2=  3,       which  is  impossible, 

b  =  1,  gives  a2  —  a  -(-1=3,    and  hence    a  :=  —  1  or  2, 

5  =  —  1,  gives  a2  +  a  -}-  1  =  3,    and  hence    a  =  1   or  —  2, 
b  =  2,  gives  a2  —  2a +  4  =  3,     and  hence    «=I, 

6  =  —  2,    gives    a~  -\-  2a  +  4  =  3>     and  hence    a  =  —  1, 
whence         a  +  &P  =  —  (1 — p),     ±  (2  -f-  p)     or     ±(i+2p). 
Similarly      c+rfp  =  ±(i— p),     ±:  (2  +  p)     or     ±(i+2p), 
and  we  have 

3=  (1—  p)(2  +  p)  =  (—  i+p)(_  2  —  p)  =  (i+2p)(—  I—  2p), 

the  proper  combinations   of   factors   being   selected  by  trial.     All   these 
factorizations  are,  however,  considered  as  identical,  since  the  factors  in 


THE   REALM    &(V 3)-  225 

each  resolution  are  associated  with  the  corresponding  factors  in  the  other 
resolutions.  All  these  factors  can  easily  be  proved  to  be  primes  of 
k(V — 3),  whence  we  see  that  3  can  be  resolved  into  the  product  of  two 
prime  factors  in  £(V — 3),  and  that  this  resolution  is  unique.  Moreover, 
all  these  factors  are  associates  of  1  —  p,  and  we  have 

3  =  — P2(i— p)2. 

We  could  have  seen  directly  from  the  equation  denning  the  realm  that 

3  =  _(V^3)2. 

Ex.  3.  If  we  endeavor  to  resolve  — 46 +  37P  into  two  factors  neither 
of  which  is  a  unit,  we  find  that  it  can  be  done  in  seven  essentially  different 
ways,  the  factors  in  each  product  not  being  associated  with  the  factors  in 
any  one  of  the  other  products. 

—  46  +  37P=  (4  +  5P)(h  +  i8p)  7) 

=  (-5  +  6p)(8  +  p)  8) 

=  (7  +  2p)(—  4  +  9P)  9) 

=  (1—  p)(—  43  —  3P)  10) 

==  (i  +  3P)(29  +  25P)  11) 

=  (4  +  3P)(5  +  22p)  12) 

=  (5  +  3P)(i  +  i7P)  13) 

We   find,   however,   that   none   of   these    factors   except    1 — p,    I+3P, 

4  +  3P>  and  5  +  3P  are  prime  numbers,   and  that  we  can  resolve  those 

which  are  not  prime  into  prime  factors  in  the  following  manner: 

4  +  5P  =  (1  —  P)  (1  +  3P),  11  +  i8p  =:  (4  +  3P)  (5  +  3P)  ; 

_5  +  6p=(i  +  3P)(4  +  3P),  8  +  p=(i-p)(5  +  3P); 

7  +  2p=z  (i_p)(4-f  3P),  _4_pc>p=  (i  +  3p)(5  +  3p); 

—  43  —  2>P  —  (i  +  3P)(4  +  3P)(5  +  3P), 

2Q  +  25P=  (1  _p)  (44- 3p)  (5  + 3p); 
5  +  22P  =  (1  —  p)  (1  +  3p)  (5  +  3p), 

i  +  i7P=  (1  —  P)(i+3P)(4  +  3P)- 

When  these  products  are  substituted  in  7),  8),  9),  10),  11),  12),  and  13) 
we  obtain  in  each  case 

—  46  +  37P=  (1— P)(i+3P)(4  +  3P)(5  +  3P)  \ 
that  is,  when  — 46 +  37P  is  represented  as  a  product  of  factors  all  of 
which  are  prime,  the  representation  is  unique.  Having  made  these  notions 
concerning  the  integers  of  £(V — 3)  clear,  we  proceed  to  what  will 
always  be  our  first  goal  in  the  discussion  of  any  realm;  that  is,  to  prove 
that  every  integer  of  &(V — 3)  can  be  expressed  in  one  and  only  one 
way  as  a  product  of  prime  numbers. 

15 


226  THE   REALM    &(V 3) 


§9.    Unique  Factorization  Theorem  for  fc(V  —  3). 

Theorem  A.  //  a  be  any  integer  of  k(\/ — 3),  and  p  any 
integer  of  &(V — 3)  different  from  0,  there  exists  an  integer  fi 
of  k{y — 3)  such  that 

n[a  —  n(3]  <n[p].1 

Let  a/fl  =  a  +  bp, 

where  a  =  r  +  rlf     b  —  s  -f-  slf  r  and  s  being  the  rational  integers 
nearest  to  a  and  b  respectively,  and  hence 

We  shall  show  that  ft,  =  r  +  Sp,  will  fulfil  the  required  condi- 
tions.    Since 

a//3  —  ix  =  r1+slP, 

n  [a/p  —  /*]  =  rt*  —  rrf,  +  st*  g  |, 

whence  n[a/p  —  fi]  <  1, 

or  multiplying  by  n[p], 

n[a  —  fxp]  <n[p]. 

The  proofs  of  the  two  remaining  theorems  which  lead  to  the 
Unique  Factorization  Theorem  and  the  proof  of  that  theorem 
itself  are  now  word  for  word  identical  with  those  in  k(i)  ;  we 
shall  therefore  merely  state  these  theorems : 

Theorem  B.  If  a  and  p  be  any  two  integers  of  £(V — j) 
prime  to  each  other,  there  exist  two  integers,  £  and  -q,  of  &(V — 3) 
such  that 

a£  +  pv=i. 

Theorem  C.  //  the  product  of  two  integers,  a  and  p,  of 
&(V — 3)  be  divisible  by  a  prime  number,  n,  at  least  one  of  the 
integers  is  divisible  by  v. 

This  theorem  has,  of  course,  the  same  corollaries  as  the  corre- 
sponding one  in  k(i). 

Theorem  i.  Every  integer  of  k(\/ — 3)  can  be  represented 
in  one  and  only  one  way  as  the  product  of  prime  numbers. 

1  See  note  in  k(i)  which  applies  equally  here. 


THE   REALM    &(V 3).  227 

§  10.    Classification  of  the  Prime  Numbers  of  &(y — 3). 

By  a  train  of  reasoning  identical  with  that  employed  in  k(i), 
it  becomes  evident  that  every  prime,  ir,  of  &(V — 3)  is  a  divisor 
of  one  and  only  one  rational  prime.  In  order  therefore  to  deter- 
mine all  primes  of  &(V — 3),  it  is  only  necessary  to  find  the 
divisors  of  all  rational  primes  considered  as  integers  of  k(  V — 3). 

Let  7T,  =a-f-  bp,  be  any  prime  of  &(V — 3)  and  p  the  positive 
rational  prime  of  which  -n  is  a  divisor. 

Then  p  =  7raJ  1) 

and  hence  p2  =  n[7r]n[a]. 

We  have  then  two  cases 

\n[a]=p,  '     \n[a]  =  i. 

i.  From  n[?r]  =inr'  =  p  and  1),  it  follows  that  a  =  irf.  From 
n  [it]  =  p  we  have  a2  —  ab  -\-  b2  =  />,  and  hence  since  every  positive 
rational  prime,  except  3,  is  of  the  form  3^  +  1  or  $n — 1,  we 
must  have,  excluding  the  case  />  =  3,  when  p  =  w[tt], 

a2 — ab-\-b2=      1,  mod  3, 
or  a2 — ab-\-b2  =  — 1,  mod  3. 

The  first  of  these  congruences  has  the  solutions 
a=      o;     a=±i;    a=i;     a== — 1] 
fr„±I;     b=      o;     6^1;     6^_i}'raod3. 

while  the  second  has  no  solutions. 

Hence  when  a  positive  rational  prime  other  than  3  is  the 
product  of  two  conjugate  primes  of  k{\/ — 5),  it  has  the  form- 
pi  +  I. 

The  case  p  =  3  is  easily  disposed  of,  for  the  equation 
p  =  a2  —  ab  +  b2  =  3 
is  satisfied  by  a=i,  b  =  —  1,  which  give 
3=(i-p)(i-p2); 

hence  3  is  the  product  of  two  conjugate  primes  of  &(y — 3). 
These  factors  of  3  are,  however,  associated,  for 

i-P2  =  -P2(i-p), 


228  THE   REALM    k(y/ 3). 

whence  3  =  —  p*-(i  —P)2,  or  3=— (V— ~3)2; 

that  is,  j,  which  is  the  only  rational  prime  divisor  of  the  discrimi- 
nant of  &(V — 3),  is  associated  with  the  square  of  a  prime  of 

ii.  From  n[a]=i  it  follows  that  a  is  a  unit.  Hence  p  is 
associated  with  the  prime  71-;  that  is,  p  is  a  prime  in  &(V — 3). 
When  p  is  of  the  form  $n —  1,  this  case  always  occurs,  for  we 
have  seen  that  in  order  to  be  factorable  in  &(V — 3)>  a  rational 
prime  must  either  be  3  or  of  the  form  311  -f-  1. 

We  shall  now  show  that  every  rational  prime,  p,  of  the  form 
3^+  1  can  be  resolved  into  the  product  of  two  conjugate  primes 

of  £(V-"3)- 
The  congruence 

x-  =  —  3,  mod  p,     />  =  3w+i, 
has  roots;  for 

(-3//0  =  (-i/7>)(3//0, 

and  if  p  =  $k-\-  1, 

(— i//>)=  1,  and  (3/70  =  (J/3), 
while,  if  £  =  46  +  3, 

(—  i//0=—  1,  and  (3//0=—  (J/3), 
and  in  both  cases  therefore 

(-3//>)  =  (/'/3)  =  (i/3)=i- 
Let  a  be  a  root ;  then 

a2  +  3  =  o,  mod  £ ; 

that  is,  (a  +  V — 3)(a — V — 3)=o,  mod />. 

Since  0+ V — 3  and  a — V — 3  are  integers  of  &(V — 3),  p 
must,  if  a  prime  in  &(V — 3),  divide  one  of  them;  we  must  have, 
therefore,  either 

g+V— 3==»       ■  I — -  2) 

when  u  and  v  are  either  both  odd  or  both  even,  or 


THE   REALM    &(\/ 3).  229 

where  u±  and  vt  are  either  both  odd  or  both  even.  But  2)  and 
3)  are,  however,  impossible,  since  ±pv=±i  implies  that  v  is 
even,  and  hence  that  p  is  a  divisor  of  1,  which  is  impossible. 

Hence  p  is  not  a  prime  in  &(V —  3),  and,  since  the  only  way  in 
which  a  rational  prime  is  factorable  in  k(  V — 3)  is  into  two  con- 
jugate primes,  p  is  factorable  in  this  manner.  The  primes  of 
&(V — 3)  may  therefore  be  classified  according  to  the  rational 
primes  of  which  they  are  factors  as  follows : 

1)  All  positive  rational  primes  of  the  form  $n -\-  1  are  factor- 
able in  &(V — 3)  into  two  conjugate  primes,  called  primes  of  the 
first  degree. 

2)  All  positive  rational  primes  of  the  form  $n  —  1  are  primes 
in  k(y — 5),  called  primes  of  the  second  degree. 

3)  The  number  3  is  associated  with  the  square  of  a  prime  of 
the  first  degree. 

It  can  be  easily  proved  as  in  the  case  of  2  in  k(i),  that  3  is  the 
only  rational  prime  which  is  associated  with  the  square  of  a  prime 
of  the  first  degree  in  &(y — 3).  We  observe  that  in  k(\/ — 3) 
as  well  as  in  k(i)  the  only  rational  primes  which  are  associated 
with  the  squares  of  primes  of  the  first  degree  are  those  which 
divide  the  discriminant  of  the  realm. 

§  11.  Factorization  of  a  Rational  Prime  in  &(V  — 3)  deter- 
mined by  the  value  of  (d/p). 

As  in  k(i),  we  can  express  the  above  results  in  a  very  con- 
venient manner  by  means  of  the  discriminant/ d,  of  &(V — 3)- 

We  have  seen  that,  when  p  —  yi-\-i,  ( — $/p)  =  i;  that  is, 
(d/p)  =  i. 

When  p  =  3,  d  is  divisible  by  p,  which  is  expressed  symbol- 
ically by  (d/p)  =0. 

Hence  we  can  classify  the  rational  primes  according  to  their  fac- 
torability  in  £(V — 3)  as  follows: 

When  (-]  =  /,     p  =  Tnrf; 

that  is,  p  is  the  product  of  two  conjugate  primes  of  the  first  degree. 

When  |-J= — j,    p  =  p; 


23O  THE   REALM    k{  V 3)- 

that  is,  p  is  a  prime  of  the  second  degree. 

When  (-\  =  o,    p  =  eir2; 

that  is,  p  is  associated  with  the  square  of  a  prime  of  the  first 
degree. 

The  primes  of  &(V — 3)  whose  norms  are  less  than  100  are  2, 
!  —  P,  5,  i  +  3p>  4  +  3Pi  5  +  3p»  5  +  6p,  7  +  3p»  7  +  6p>  5  +  9p> 
7  +  9/°,  i  +  9p»  io  +  3p,  n  +  3p. 

§  12.    Cubic  Residues. 

If  a  and  m  be  rational  integers  and  a  be  prime  to  m,  a  is  said 
to  be  a  cubic  residue  or  non-residue  of  m  according  as  the 
congruence 

x3  =  a,  mod  m, 
has  or  has  not  roots. 

As  in  the  development  of  the  theory  of  biquadratic  residues, 
we  saw  that  our  field  of  operation  must  be  not  simply  the  rational 
integers  but  the  integers  of  the  realm  k(i),  of  which  the  rational 
integers  are  a  part,  so  in  the  theory  of  cubic  residues  we  must  take 
as  our  field  of  operation  the  integers  of  &(V — 3)  ;  that  is,  we 
must  consider  the  congruence 

x3zz=a,  mod  /a, 


where  a  and  /x  are  integers  of  k{  V  — 3)  and  a  prime  to  /x. 

Lack  of  space  forbids  even  a  brief  discussion  of  this  subject 
here  but  the  reader  should  consult  Bachmann:  Die  Lehre  von 
der  Kreistheilung,  I4te  Vorlesung ;  Jacobi :  Works,  Vol.  6,  p.  223, 
and  Eisenstein :  Crelle,  Vols.  27  and  28. 


CHAPTER  VII. 
The  Realm  k(y/2). 

§  i.    Numbers  of  &(\/2). 
The  number  V2  is  defined  by  the  equation 
x2  —  2  =  0, 

which  it  satisfies.  All  numbers  of  k{  V2)  have  the  form  a  -\-  by  2, 
where  a  and  b  are  rational  integers. 

The  other  root, — \^2,oi x- — 2=0  defines  the  realm  k{ — V2)> 
conjugate  to  &(V2)-  The  two  realms  are,  however,  as  in  both 
the  previous  cases,  identical. 

The  conjugate  of  a,  —  a-\-by2,  is  a',  =  a —  by 2.  The 
product  act!  is  called  as  before  the  norm  of  a  and  is  denoted  by 
n[a]. 

In  n[a]  =  (a-\-by2)(a —  by 2)=  a2  —  2b2  we  notice  the 
first  of  a  series  of  important  differences  between  this  realm  and 
k(i)  and  k(y — 3)-  The  norm  of  a  number  of  k(y2)  is  not, 
as  heretofore,  necessarily  a  positive  rational  number.  It  may  be 
either  a  positive  or  negative  rational  number.  This  will  easily  be 
seen  to  be  true  of  all  quadratic  realms  defined  by  real  numbers, 
while  the  norms  of  numbers  of  quadratic  realms  defined  by 
imaginary  numbers  -are  always  positive.  Realms  of  the  first 
kind,  as  k(  V2)>  are  called  real  realms;  those  of  the  second  kind, 
as  k(i)  and  &(V — 3),  imaginary  realms. 

We  have  evidently ;w [a/?]  =n[a]n[j3],  where  a  and  /?  are  any 
numbers  of  &(\/2).   . 

§2.    Integers  of  k {^/.2 ). 

Writing  all  numbers  of  k(  V2)  *n  tne  form 

a  =  <h  +  biV2 

where  alf  blf  c1  are  rational  integers,  having  no  common  factor, 

231 


232  THE   REALM    k(\/2). 

we  can  show  exactly  as  in  k(i)  that  a  necessary  and  sufficient 
condition  for  a  to  be  an  integer  is  ^=1. 

Therefore  all  integers  of  k(\/2)  have  the  form  a-\-b\/2, 
where  a  and  b  are  rational  integers,  and  all  numbers  of  this  form 
are  integers;  that  is,  1,  V2  is  a  basis  of  k{  V2)- 

§3.    Discriminant  of  ^(V2)  • 

The  discriminant  of  &(V2)  is  the  squared  determinant 

1.1  V~2\* 

|l       -l/2| 
formed    from   a   pair   of   basis   numbers   and   their    conjugates. 
Denoting  it  by  d,  we  have 

d=8. 
§  4.     Divisibility  of  Integers  of  k(  V2)- 

The  definition  is  identical  with  that  given  in  R,  k(i)  and 
&(V — 3).     For  example,  since 

14  +  9V2=  (2  +  V2)  (5  +  2V2) 

14  +  9V2  is  divisible  by  2  +  2V2  and  5  +  2\/2. 

On  the  other  hand,  since  no  integral  values  of  x  and  y  exist  for 
which  the  equation 

5  +  2  V2  =  ( 1  +  2  V2)  O  +  y  V2) 

is  satisfied,  5  +  V2  is  not  divisible  by  1  +  2  V2- 

§  5.    Units  of  k ( V2) •    Associated  Integers. 

The  units  of  &(V2)>  being  those  integers  of  k(\/2)  which 
divide  every  integer  of  the  realm,  divide  1,  and  since  all  divisors 
of  1  are  evidently  units,  they  can  be  defined  either  as  those 
integers  of  k{^2)  whose  norms  are  either  1  or  — 1,  or  as  those 
integers  of  k(  -\f  2)  whose  reciprocals  are  also  integers. 

Let  €,  =  x  -f.  3/y  2,  be  a  unit  of  &(V2)  ;  we  have  then  either 

n[c]  =  i,  or  n[e]  = — 1; 
that  is  i.  x2  —  2y2=i,  or  ii.  x2  —  2y2  =  — I.1 

1  The  reader  will  recognize  i  and  ii  as  special  cases  of  Pell's  Equation 

x2  —  Dy2=±i, 

a  discussion  of  which  will  be  found  Chap.  XIII,  §  5.     Here  we  shall  treat 
the  question  from  a  different  point  of  view. 


THE   REALM    &(V2)-  233 

We  can  easily  obtain  many  solutions  of  both  i  and  ii,  as,  for 
example : 

\v=±    i,  y=        o,  c=±  1, 
x=±    7,,y=±    2,  £=±3±2V2, 
x=±l  17,  y=  ±  12,  €=±  17  ±  l2\/2, 

x=  ±    1,  y=  ±    I,  €=±l±  V2, 
<*===£   7,  :y=±   S,'.€=±:7.±5V3r, 

$•  =  ±  41,  y=  ±29,  c=  ±  41  ±  29 \/2. 

We  shall  now  show  that  &(V2)  has  indeed  an  infinite  number 
of  units,  each  of  which  can,  however,  be  represented  as  a  power 
of  the  unit  I  +  \/2,  multiplied  by  -f- 1  or  —  i.  This  unit  I  +  V2 
is  called  the  fundamental  unit. 

Theorem  i.  All  units  of  k{^/2)  have  the  form  ±{i  +  V2)n> 
•where  n  is  a  positive  or  negative  rational  integer  or  o,  and  all 
numbers  of  this  form  are  units  of  k(y/2). 

Let  e=i  +V2-  We  see  that  every  positive  power  of  c  is  a 
unit;  for 

»[c»]  =  (»[€])»=( — i)*=i  or  — 1. 

Hence  en  is  a  unit. 

Moreover,  since  €"£-w=i, 

e""  is  a  unit  also;  that  is,  all  negative  powers  of  c  are  units, 
Furthermore  two  different  positive  powers  of  c  give  always  dif- 
ferent units;  for,  since  c,  =  I  +V2>  is  greater  than  i,  the  positive 
powers  of  c  are  all  greater  than  I  and  continually  increase.  Hence 
no  two  are  equal. 

Also,  since  €-*  =  i/<pt 

it  is  evident  that  c-1  is  less  than  I  and  hence  that  the  negative 
powers  of  c  are  all  less  than  I  and  continually  decrease.  There- 
fore no  two  negative  powers  are  equal,  and  no  negative  power  is 
equal  to  any  positive  power.  Hence  every  power  of  e  is  a  unit 
of  &(V 2),  and  two  different  powers  give  always  different  units. 
Therefore  k(  V2)  possesses  the  remarkable  property  of  having 
an  infinite  number  of  units.    We  shall  now  show  that  the  powers 


234  THE   REALM    &(V2). 

of  c  multiplied  by  ±  I  are  all  the  units  of  k( \/2)  ;  that  is,  if 
r)  be  any  unit  of  k(  V2)>  it  will  be  of  the  form 

where  n  is  positive,  negative  or  o. 

Let  0  +  &V2  De  any  unit  °f  &(V2)-  Then  a  —  b  V  2> 
—  a  +  &V2  and  — a  —  ^V2  w^  also  be  units  of  k(  V2)-  Denote 
that  one  of  these  four  units  which  has  both  terms  positive  by 
-qx{b  may  be  o),  the  remaining  three  will  be  — -qx,  rj^  and  — •>//. 
We  shall  show  that 

where  n  is  positive  or  o. 

Since  Vi^1, 

it  follows  that  7j1  =  en, 

Or  €n<77l<€TC+1  i) 

where  n  is  a  positive  integer  or  o.     We  shall  show  that  the  latter 
case  can  never  arise.     Dividing  i )  by  cn,  we  have 

i  <  wfe  <  €, 

where  ^/c*  is  a  unit,  since  the  quotient  of  two  units  is  a  unit. 

Let  r)1/en  =  x-\-y\/2. 

We  have  (x-\-y^2)(x  —  y^/2)  =  ±I, 

and  hence,  since  x  +  yV2  >  x>  it  follows  that 

\x  —  yV2l  <i; 

that  is  — i  <  x  —  y  V2  <  J- 

This  combined  with 

i  <^  +  yV2<  i  +  V2  2) 

gives  o  <  2x  <  2  -f-V2> 

and  hence,  ^  being  a  rational  integer, 

jr=  i. 
But,  if  4'=  i,  it-  is  evident  that  no  rational  integral  value  of  y 
will  satisfy  2),  for  positive  values  of  y  give 

i  +  yV^i+V2, 

and  y  =  o,  or  a  negative  integer  makes 


THE   REALM    k(^/2).  235 

I  +3>y2<  1. 
Hence  i)  is  impossible,  and  we  have 

and  therefore  — Vi==  —  c"5 

and  since  77^/  =  ±  1, 

77/  =  ±  i/cn  =  ±  cn,  and  —  q/==  zp  rn. 

Therefore,  if  77  be  any  one  of  the  four  units  77^  — rjlf  77/,  — 17/, 
that  is  any  unit  of  &(V2)>  we  have 

where  n  is  positive,  negative  or  o. 

We  can  express  all  units  of  k{i)  in  the  form  in,  but  obtain  only 
the  four  different  ones  i,i,  —  1,  — i,  since  *4  =  l. 

Likewise  we  can  express  all  units  of  &(V — 3)  in  the  form 
±pn,  but  obtain  only  the  six  different  ones  1,  — 1,  p,  — p,  p-, 
—  p2,  since  p8  =  i. 

Any  two  integers  which  differ  only  by  a  unit  factor  are  said  to 
be  associated,  and  in  all  questions  of  divisibility  are  considered  as 
identical.  Thus,  if  a  be  a  factor  of  /x,  and  n  any  positive  or 
negative  rational  integer,  the  infinitely  many  integers  ±  ena,  that 
are  associated  with  a,  are  also  factors  of  p..  All  these  factors, 
however,  are  considered  as  the  same.  With  this  understanding, 
we  shall  find  that  the  fact  that  k(  V2)  nas  an  infinite  number  of 
units  in  no  way  interferes  with  our  adopting  definitions  for  prime 
and  composite  numbers  of  k{  \/2)  identical  with  those  used  in  the 
previous  realms  and  proving  the  unique  factorization  theorem  for 
the  integers  of  &(\/2). 

§  6.    Prime  Numbers  of  k  ( y/2 ) . 

The  definitions  are  identical  with  those  in  the  preceding  realms 
and  we  can  determine  whether  an  integer  is  prime  or  composite 
by  the  methods  employed  in  those  realms. 

Ex.  1.    To  determine  whether  13  +  12V2  is  prime  or  composite. 

Put  13  +  i2V2=  (a  +  b V2)  {c  +  d^2)  ; 

then  —119=  (a2  —  2b2)  (c2  —  2d2). 


236  THE   REALM    &(\/2). 

There  are  only  four  distinct  cases  to  be  considered 


J  a~  —  2b-  =17,  ..  j  a2  —  2fr2  =  —  17, 

|  c2  —  2d3  =  —  7.  J  c'2  —  2d2  =  7. 

.....  f  a2  —  2b~  =  ±  119, 

111  and  iv.         ^ 

\c2  —  2di=±i. 

Both  iii  and  iv  give  c  -f-  rf\/2  a  unii  and  therefore  need  not  be  considered. 
As  solutions  of  i  we  have 

a  =  ±  5,      b=±2,      c  =  ±  1,      d=±2, 
which  give 

I3  +  I2\/2=  (5  +  2V2)(l+2y2)  =  (_5_2V2)(— I— 2V2), 

the  proper  factors  being  selected  by  trial. 

Since  neither  of  the  integers  S+V2»  I+2y'2  is  a  unit,   13  +  12^2 
is  a  composite  number. 

Other  solutions  of  i  are 

a  =  ±7,      b  =  ±4,      c=±u,      d  =  ±8, 
which  give 

i3  +  i2V^=(7-4V2)(n  +  8V2)  =  (-7  +  4V2)(-II-8V2)- 
As  solutions  of  ii  we  have 

a  =  ±i,      b  =  ±3,      c  =  ±5,      d  =  ±3, 
which  give 

13  +  12^2  =  (— 1+  3^2)  (5  +  3V2)  =  (I—  Z\J~2)i—  5  —  3\/2)- 
We  see,  however,  that  all  these  factorizations  can  be  derived  from  any 
particular  one  by  multiplying  the  factors  by  suitable  units,  and  hence  are 
not  different;  that  is, 

7-4^/2  =  e-2(5  +  2y2),     ii+8y2  =  e2(i+2y2), 
—  1  +  3^2  =  r*(s  +  2^2),        5  +  3 V2  =  c  C*  +  2V2)' 
where  e=  1  +V2»  anc*  we  nave  *n  general 

13  +  12V2  =  [±  e«(5  +  2V2)  ]  [±  r*(i  +  2V2)  ]• 
Ex.  2.     Prove  that  \-\-2-\j2  is  a  prime. 
§7.    Unique  Factorization  Theorem  for  fc(V2)- 
Theorem  A.     //  a  be  any  integer  of  k(  ^ 2),  and  (3  any  integer 
of  &(V<?)  different  from  0,  there  exists  an  integer  /a  of  k(^2) 
such  that 

\n[a-rf]\<\n[p]\1 
Let  a/(3  =  a-\-b-\/2, 

1  See  note  to  corresponding  theorem  in  k(i)  which  applies  equally  here. 


THE   REALM    k(\^2).  237 

where  a  =  r-\-r1,  b  =  s-\-s1,  r  and  ^  being  the  rational  integers 
nearest  to  a  and  b  respectively,  and  hence 

We  shall  show  that^u,  =  r -{- s\/2,wi\\  fulfil  the  required  condi- 
tions.    Since 

d/p—ti=rx  +  st-s/2t 
\n[a/p  —  fi]  I  =  \r*  r-  2S* \  ^  J, 
whence  \n[a/P  —  /*]  |  <  i, 

or,  multiplying  by  \n[/3]  |, 

|n[a  — ^]|  <|n[0]|. 

The  proofs  of  the  two  theorems  which  lead  to  the  unique  factori- 
zation theorem  and  that  of  the  unique  factorization  theorem  itself 
are  identical  with  those  in  k(i)  and  fc(V — 3)  with  the  exception 
that  the  absolute  value  of  the  norm  is  substituted  for  the  norm  of 
an  integer.  This  is  evidently  necessary  whenever  we  make  a 
comparison  between  two  integers  of  k(  V2)  similar  to  that  made 
between  rational  integers  when  we  say  that  one  is  greater  in 
absolute  value  than  the  other.  It  is  also  necessary  when  we  ex- 
press the  result  of  an  enumeration  as  a  function  of  an  integer  of 
k(\/2).  In  k(i)  and  fc(V — 3)  tne  norms  of  all  numbers  were 
positive  and  hence  were  their  own  absolute  values. 

The  result  of  an  enumeration  being  always  a  positive  integer, 
the  conception  of  the  positive  integer  being  indeed  arrived  at  by 
considering  it  as  representing  the  result  of  an  enumeration,  to 
express  such  a  result  as  a  function  of  an  algebraic  integer,  a,  we 
must  have  some  function  of  a  which  is  always  a  positive  integer. 
Such  a  function  is  |w[a]|. 

Theorem  B.  //  a  and  (3  be  any  two  integers  of  k{^/2)  prime 
to  each  other,  there  exist  two  integers,  $  and  r),  of  k( y<?)  such  that 

Theorem  C.  //  the  product  of  two  integers,  a  and  /?,  of 
&(V<?)  be  divisible  by  a  prime  number,  w,  at  least  one  of  the 
integers  is  divisible  by  », 

Theorem  2.  Every  integer  of  k(\/2)  can  be  represented  in 
one  and  only  one  way  as  the  product  of  prime  numbers. 


238  THE   REALM    k(\/2). 

§  8.    Classification  of  the  Prime  Numbers  of  k ( y  2) . 

By  a  train  of  reasoning  identical  with  that  employed  in  the 
preceding  realms,  it  becomes  evident  that  every  prime,  n,  of  k(\/2) 
is  a  divisor  of  one  and  only  one  rational  prime.  In  order  there- 
fore to  obtain  all  primes  of  &(V2)  it  is  only  necessary  to  resolve 
all  positive  rational  primes  considered  as  integers  of  &(V2)  into 
their  prime  factors  in  that  realm. 

Let  7r,  =  fl-f-  frV 2.  be  any  prime  of  k(^/2)  and  p  the  positive 
rational  prime  of  which  it  is  a  divisor. 

Then  £= jtO^  I ) 

and  hence  />2-=E=^I^l^I^]  • 

We  have  then  two  cases 

n[a\j==JK  \n[a]=  1. 

i.  From  n[y]  =  Tnr'  —  p  and  1)  it  follows  that  a  =  7r'. 
Since  every  positive  rational  prime,  except  2,  is  of  one  of  the 
forms  Sn  ±  1,  8n  ±  3,  we  must  have  (excluding  the  case  p=2)9 

when  £  =  »|y], 

a2  —  2&2=      1,  mod  8,  2) 

or  a2  —  2&2  =  — 1,  mod  8,  3) 

or  a2  —  2b2  ==      3,  mod  8,  4) 

or  a2  —  2&2  =  — 3,  mod  8.  5) 

The  first  of  these  congruences  has  the  solutions 

as*  ±ii     ±1,     ±3,     ±3 


,  mod  8. 
b  =  ±  2,         o,     ±2, 

The  second  has  the  solutions 

11-4:1.     ±i,     ±3,     ±3)        odg 
&  —  =fci,     ±3.     ctii     ±3J 

The  last  two  have  no  solutions,  for  they  give 

a2^2&2  ±  3,  mod  8, 

and  hence  require  that  2b2  ±  3  shall  be  a  quadratic  residue  of  8. 
But  the  only  quadratic  residues  of  8  are  J  and  4,  whence  it  follows 


THE   REALM    k(^/2).  239 

that  a  necessary  condition  that  3)  or  4)  shall  have  a  solution  is 
1  ==2b2  ±  3,  mod  8,  or  4  =  2fr2  ±  3,  mod  8. 

All  four  of  these  congruences  are  easily  seen  to  have  no  solu- 
tions, and  4)  and  5)  therefore  have  no  solutions. 

Hence  when  a  positive  rational  prime  other  than  2  is  the  prod- 
uct of  two  conjugate  primes  of  k(\/2),  it  has  the  form  8n±  1. 

The  case  p  =  2  must  next  be  considered. 
The  equation  a2  —  2b2  =  2 

is  satisfied  by  a  =  ±2,     &  =  ±i. 

Hence  2=  (2  +  yi)  (2—  yi)  =  (1  +  V2)  (—  I  +  V~2)  ( V2)2 ; 
that  is,  2,  which  is  the  only  rational  prime  divisor  of  the  dis- 
criminant of  &(V 2)  is  associated  with  the  square  of  a  prime  of 

ii.  Since  n[a\=  1,  a  is  a  unit.  Hence  p  is  associated  with  the 
prime,  ?r;  that  is,  p  is  a  prime  in  k(\/2).  When  p  is  of  the  form 
8n  ±  3  this  case  always  occurs,  for  we  have  seen  that  to  be  fac- 
torable in  k{^2)  a  rational  prime  must  either  be  2  or  of  the  form 
Sn  ±  1. 

We  shall  now  show  that  every  rational  prime,  p,  of  the  form 
8n  ±  1  can  be  resolved  into  the  product  of  two  conjugate  primes 
of  fc(Vl). 

The  congruence  x2  =  2,  mod  p,  p  =  8n±i,  has  roots,  for 
(2/p)  =  i  when£  =  8w±i. 

Let  a  be  a  root ;  then 

a2  =  2,  mod  />; 

that  is  (fl+V2)(fl — V2)— °>  m°d  />• 

Since  a  +V2  and  a — \/2  are  integers  of  &(V2),  />,  if  a  prime 
of  k(\/2),  must  divide  either  a  +V2>  or  a —  V2-  This  is,  how- 
ever, impossible,  for  from 

a  ±y2  =  p(c  +  dy2), 
where  c-\-d^/2  is  an  integer  of  &(V2)>  ft  would  follow  that 

pd=±  1, 
which  is  impossible,  since  £  and  d  are  both  rational  integers  and 
/>  >  1.     Hence  £  is  not  a  prime  in  &(  V2),  and  since  the  only  way 


240  THE   REALM    k(\/2). 

in  which  a  rational  prime  can  be  factored  in  &(\/2)  is  into  two 
conjugate  prime  factors,  p  is  factorable  in  this  manner. 

The  primes  of  k{  V2)  may  therefore  be  classified  according  to 
the  rational  primes  of  which  they  are  factors  as  follows : 

i)  All  positive  rational  primes  of  the  form  8n  ±  I  are  factor- 
able in  £(y<?)  into  two  conjugate  primes,  called  primes  of  the 
first  degree. 

2)  All  positive  rational  primes  of  the  form  8n  ±  3  are  primes 
in  k{ y 2),  called  primes  of  the  second  degree. 

3)  The  number  2  is  associated  zvith  the  square  of  a  prime  of 
the  first  degree  in  k{  ^ 2). 

It  can  be  shown,  as  in  the  cases  of  2  in  k(i)  and  3  in  &(  V — 3), 
that  2  is  the  only  rational  prime  that  is  associated  with  the  square 
of  a  prime  of  the  first  degree.  We  observe  that  2  is  the  only 
rational  prime  divisor  of  the  discriminant. 

§  10.  Factorization  of  a  Rational  Prime  in  k{  y^)  determined 
by  the  value  of  (d/p). 

As  in  k(i)  and  &(y — 3),  the  above  results  can  be  expressed 
in  tabular  form  by  means  of  the  discriminant  of  k(~\/2).  The 
formation  of  such  a  table  will  be  left  to  the  reader. 

§  11.     Congruences  in  k{^/2). 

The  unique  factorization  theorem  having  been  proved  for  the 
integers  of  k(^2),  a  series  of  theorems  analogous  to  those 
deduced  in  the  case  of  the  preceding  realms  can  be  shown  to 
hold  for  the  integers  of  k(\/2). 

Having  defined  the  congruence  of  two  integers  of  k(^/2)  with 
respect  to  a  modulus  precisely  as  we  defined  that  of  two  rational 
integers,  we  should  find  that  there  are,  with  respect  to  a  given 
modulus  fi,  I  11  [fx]  I  classes  of  incongruent  numbers,  and  can  then 
deduce  for  the  integers  of  k(\^2)  Fermat's  theorem  and  other 
theorems  relating  to  congruences. 

§  12.    The  Diophantine  Equations 
x2  —  2y2  =  ±i,  x2  —  2y2=±p,  and  x2  —  2y2=±m.1 

It  is  required  to  find  the  rational  integral  values  of  x  and  y 
1  See  Chap.  XIII,  §  5. 


THE   REALM    &(\/2).  24  I 

for  which  these  equations  are  satisfied.  Since  the  first  member 
of  each  of  the  equations  is  the  norm  of  x  -f-  y  V  2,  the  problem 
reduces,  in  the  light  of  what  we  have  learned  about  the  integers 
of  &(V2),  to  that  of  finding  an  integer  of  k{  V2)  whose  norm  is 
the  quantity  constituting  the  second  member  of  the  equation. 
If  a  -f-  &  V2  be  such  an  integer,  then 

x  =  ±  a,    y=±b, 

evidently  satisfy  the  equation  under  consideration.  We  see  also 
that,  if  any  one  of  these  equations  has  a  single  solution,  it  has  an 
infinite  number  of  solutions,  for  if  x  =  a,  y  =  b  be  a  solution  of 
the  given  equation,  and 

(a  +  b\^2)e2n  =  a1  +  bxy~2, 
where  e=  I  +V2>  and  n  is  any  positive  or  negative  integer  or  o, 
then  since 

wlA  +  ^V2]  =n[(a-\-b\/2)e2n]  =n[a-j-&\/2], 
x  =  alty  =  bx  is  also  a  solution  of  the  given  equation.  Moreover, 
since  no  two  powers  of  c  are  equal,  the  solutions  obtained  by 
giving  n  any  two  different  values  are  different.  Hence  the  solu- 
tions are  infinite  in  number.  We  shall  consider  now  each  of  the 
equations  in  detail. 

i.  x2  —  2y2=i,  ii.  x2  —  2y2  =  — I. 
The  necessary  and  sufficient  condition  that  an  integer  of 
£(V2)  shall  have  the  norm  db  I  is  that  it  shall  be  a  unit.  All 
units  having  the  norm  I  are  included  in  the  form  ±(i  +V2)2n> 
and  all  having  the  norm  —  i  in  the  form  ±(i  -|-V2)2n+\  n  being 
a  positive  or  negative  integer  or  o.  Negative  values  of  n  repeat 
solutions  given  by  positive  values,  since  (i+V2)""  1S  the  con- 
jugate of  (i+V2)n-    Hence,  if 

±{i+V2)2n  =  a  +  by2, 

x=±a,     y  =  ±b, 
satisfy  i,  and  if 

±(i+V2)2n+1  =  a1  +  b1V~2} 

x=±alf     y  =  ±b1, 

satisfy  ii,  and  these  are  all  the  solutions  of  i  and  ii. 
16 


242  THE   REALM    k{^2). 

For  example: 

±(i+V2)2=±  (3  +  2V2)  gives  (±3)2  —  2(±2)2=i; 
that  is  x  =  ±  3 ;  3'  =  ±  2  are  solutions  of  i ; 

while 

±  (i+V2)3=±  (7  +  5V2)   gives   (±7)2  — 2(±5)2  =  — 1; 
that  is  x=±7;  y=  ±  5  are  solutions  of  ii. 

iii.  x2 —  2y2  =  p,     iv.  x2  —  2y2  =  —  p, 
where  p  is  a  positive  rational  prime.     The  necessary  and  sufficient 
condition  that  ±  p  should  be  the  norm  of  an  integer  of  &(\/2)  is 
p  =  ±  1,  mod  8,  or  p  =  2.     Hence  iii  and  iv  are  solvable  when 
and  only  when 

^±1,  mod  8,  or  p  =  2. 

Let  p=±  1,  mod  8. 

If  x  =  a}  y  =  b  be  any  solution  of  iii,  all  rntegers  of  the  form 
(a  ±  b^2)e2n  =  a1  +  b{\/~2 
give  solutions  of  iii,  x=  ±aly  y=±b1;  for 

n[(a  ±  b^2)e2n]=n[a  ±  b^/2]  (—  i)2n  =  p, 
and  all  integers  of  the  form 

( a  ±  b y 2 ) €2n+1  =  a2-\-b2y2 
give  solutions  of  iv,  x=±a2,  y=±b2;    for 

n[(fl±^V2)€2"+1]  =n[a±:  b^2](—i)2n+1  = —  p. 
These  are  easily  seen  to  be  all  of  the  solutions  of  iii  and  iv. 
Ex.  1.    To  find  all  rational  integral  solutions  of  the  equations 
x2  —  2y2  =  7,       x2  —  2y2  —  —  7. 
A  solution  of  the  first  equation  is 

Hence   (3  ±V2)  (J  +V2)2nr  gives  all  solutions  of  the  first  equation  and 
(3±V2)(i  +\/2)2n+1   all   solutions   of  the   second. 
Thus  for  example 

(3  +  V2)  (1  +  \Z2)2=  13  +  9V2      : 

(3-  V2)  (1  +  V2)*  =  5  +  3V2 

(3  +  V2)  ( 1  +  \/2)  =  5  +  4V2" 

(3  —  V2)  (1  +  V2")  =  1  +  2V2" 

v.  x2  —  2y2  =  m,     vi 


gives 

(±i3)2-2(±9)2  =  7, 

gives 

(±5)2-2(±3)2  =  7, 

gives 

(±5)2-2(±4)2  =  - 

gives 

(±l)2-2(±2)2  =  - 

i.    X2- 

-  2y2  =  —  m, 

THE   REALM    &(V2)-  243 

where  m  is  a  positive  rational  integer.  Since  m  must  be  the  norm 
of  an  integer  of  k(\/2),  and  hence  must  be  factorable  into  two 
conjugate  integers  of  k(y/2),  the  necessary  and  sufficient  condi- 
tion that  v  and  vi  shall  have  solutions  is  that  every  rational  prime 
factor,  p,  of  m  such  that  £ss  ±.  3,  mod  8,  shall  occur  to  an  even 
power. 

If  then  m  =  Pip2  ' '  •  Prqx2tiq22t2  ■  ■  ■  q*2t', 

where  Pi,P2>  "  -,Pr  =  ±  1,  mod  8,  or  =  2, 

and  qlfq2,  ...,^=±3,  mod  8, 

we  have 

m={irxiz2  ■  •  •  Trrq^q^  •  •  •  g,*«)  (77-/77-./  •  ■  •  wr'q%**q2u  ■  ■  •  £,**)»  0 
=  (a  +  &V2)  (a  —  &V2)  =a2  —  2b2, 
and  #==  ±  a,  y  =  ±  &  are  solutions  of  v.  If  we  interchange  any 
77-  in  one  factor  of  1)  with  its  conjugate,  we  shall  obtain  a  different 
factorization  of  m  unless  «[tt]  =  2,  in  which  case  the  factoriza- 
tion is  not  different,  since  the  factors  of  2  are  identical. 

Suppose  this  interchange  of  in  and  tt/,  m[tt]  =f=2,  to  have  been 
made,  giving 

m=  (ax-\-  &iV2)  (ai  —  ^iV2)  =«i2  —  2&x*. 
Then  x  =  ±  aly  y  =  ±  bx  are  new  solutions  of  v.  Suppose  that 
by  these  interchanges  of  one  or  more  7r's  with  their  conjugates  we 
obtain  all  possible  different  factorizations  of  m.  Then  by  multi- 
plying a  factor  of  each  of  these  factorizations  by  the  even  powers 
of  e  in  turn  we  obtain  from  each  factorization  an  infinite  number 
of  solutions  of  v,  and  by  multiplication  by  the  odd  powers  of  e 
in  turn  we  obtain  from  each  factorization  an  infinite  number  of 
solutions  of  vi,  and  these  are  all  the  solutions  of  v  and  vi.  That  is, 
if  a-i  +  b± V2,  a2  +  &2  V2,  •  •  •  ,  at  +  bt  V2 

be  each  a  factor  of  a  different  one  of  the  t  factorizations  of  m,  all 
solutions  of  v  are  given  by 

(di±  bi^2~)e2n=Cin  f  C?*nV2, 
whence  x  =  ±  r«  ,     y  =  ±  </»n, 

and  all  solutions  of  vi  are  given  by 

(a*  ±  ^V2)e2n+1=^^n  +  /iV2, 


244 


THE    REALM    k{\/2) 


whence  x=±ein,    y=±fin, 

where  i=  1,2,  •••,  t,  andn  =  o,  1,  ••  •  . 

Ex.  2.    To  find  all  rational  integral  solutions  of  the  equations 


x'  —  2y  =  1 19 


2r  =  — 119. 


and      x~ 
We  have 

119  =  7  .  17  =  (3  +  y 2")  (3  —  y  2)  (5  +  2^2)  (5  —  2y 2) 

=  [(3  +  V2)(5  +  2y2)][(3-y2)(5-2y2)] 

=  (i9+ny2)(i9— ny2), 
or  =[(3+y2)(5_2y2~)][(3-y2)(5  +  2y2)] 

=  (n_y2)(ii+y2"). 

Whence    we    see    that    (19  ±  nV2)e2«    and    (11  ±  V2)e2»    gjve    au    the 
solutions  of  the  first  equation,  and  (19  ±  ii\/2)e2n+i  anci  (IX  ±  y2)e2n+i 
give  all  the  solutions  of  the  second. 
Thus,  for  example: 

(i9+ny2~)(i  +  y2)  =4i  +  3oy2"   gives  (±4i)2  — 2(±3o)2  =  — 119, 

(19—  ny2)(i  + V2)  —  —  3  +  8V2"  gives  (±  3)2_2(±  8)2=:  —  119, 

(11  +  y2~)(i  +  y2)  =  i3  +  i2y^    gives  (±  n)2  —  2(±i2)2  =  —  119, 

(11  —  V2)(i  +  y^)  =9  +  10V2"      gives  (±9)'-  —  2(±  io)2z=  —  119. 


CHAPTER  VIII. 
The  Realm  &(V — 5). 

§  1.    Numbers  of  &(V— 5)  -1 

The  number  V —  5  ls  defined  by  the  equation 

that  it  satisfies.  All  numbers  of  &(V — 5)  have  the  form 
a  -f-  &V —  5,  where  a  and  b  are  rational  numbers. 

The  conjugate  of  a,  =  a-\-by — 5,  is  a',  =a  —  by — 5;  also 

4a]=a2  +  5&2, 
and  n[a/3]=n[a]n[l3] 

§2.    Integers  of  k ( V — 5). 

Writing  all  numbers  of  &(y — 5)  in  the  form 

fli  +  K  V^ 
a  = — , 

where  alt  blf  cx  are  rational  integers,  having  no  common  factor, 
we  can  show  exactly  as  in  k(i)  that  a  necessary  and  sufficient 
condition  for  a  to  be  an  integer  is  ^=1. 

Therefore  all  integers  of  fc(V — 5)  nave  the  form  a  +  b V — 5 
where  a  and  b  are  rational  integers,  and  all  numbers  of  this  form 
are  integers;  that  is,  1,  V — 5  is  a  basis  of  &(V — 5). 

§  3.    Discriminant  of  k(  V —  5 )  • 

The  discriminant  of  k(\/ — 5)  is 

1 1,      v-s2 

i  =  —  20. 

\i,    -V-S] 
§4.     Divisibility  of  Integers  of  &(V  —  5). 

The  definition  is  identical  wTith  that  adopted  heretofore. 

1  Throughout  this  chapter  see  corresponding  sections  in  k(i). 

245 


246  THE   REALM    &(V 5). 

Ex.  I.     We  see  that  1  -+-  5\/ —  5  is  divisible  by  2  -f-  >/ —  5,  since 

1  +  JSV^  =  (2  +  V- 5)  (3  +  V- 5)  • 
Ex.  2.    We  see  that  5  +  2 V —  5  is  not  divisible  by  4  -f-  V —  5,  since 
5  +  2V:=r5=  (4  +  n/-7?)  O  +  W^) 
holds  for  no  integral  values  of  x  and  y. 

§5.    Units  of  fe(V—  5).     Associated  Integers. 

The  units  of  &(y — 5)  are  defined  as  were  those  of  the  pre- 
ceding realms,  and  as  the  norm  of  a  number  of  &(V — 5)  is 
always  positive,  the  necessary  and  sufficient  condition  that 
c,  =  x  -f-  y  V —  5,  shall  be  a  unit  is 

n[e]=x*  +  5y*  =  i, 
which  gives  y  =  o,     jt  =  ±  1 . 

Hence  1  and  —  1  are  the  units  of  k{  V — 5). 

The  definition  of  associated  integers  and  the  conventions  re- 
garding them  are  identical  with  those  heretofore  adopted ;  that  is, 
the  integers  a  and  —  a,  obtained  by  multiplying  any  integer  a  by 
the  units  1  and  —  1,  are  said  to  be  associated,  and  in  all  questions 
of  divisibility  are  considered  identical. 

§6.    Prime  Numbers  of  &(V  —  5). 

The  definitions  are  identical  with  those  in  the  preceding  realms. 

Ex.  1.     To   determine  whether  2  is  a  prime  or  composite  number  in 

Put  2  =  U  +  y V=5)  («  +  *V— 5)  J 

then  4  =  (V  +  5V2)  (w2  +  $v2), 


fjra  +  5y2  =  2  (  x2 

(„*  +  5^  =  2        °r"-     {«■ 


and  hence 

Evidently  i  is  impossible  since  x  and  y  must  be  rational  integers. 
From  ii  it  follows  that  w  +  v V —  5  is  a  unit.     Hence  2  is  a  prime  in 

*(V=5). 

Ex.  2.    To  determine  whether  1  -J-  >/ —  5  is  a  prime  or  composite  num- 
ber of  KV-7^)- 

Put  1  +  \/=5=  U  +  y \/^S)  («  +  ^V— 5) ; 

then  6=  {x2  +  5y2)  (w2  -f  5zr), 


THE   REALM    fc(V 5)-  247 

and  hence 

i-     <     o  ..  or  11.     I    , 

\  u2  +  sir  —  2  \  u2  +  sir  =  1 

from  which  it  is  evident  as  above  that  1  +  V — 5  is  a  prime  in  k  ( >/ —  5) . 

We  observe  that  we  have  in  i-f-  V —  5  the  first  instance  of  a 
prime  number  whose  norm  is  not  a  power  of  a  rational  prime. 

We  shall  see  later  that  a  necessary  and  sufficient  condition  for  the  norms 
of  all  complex  primes  of  any  given  quadratic  realm  to  be  rational  primes  is 
that  the  unique  factorization  theorem  shall  hold  for  the  integers  of  the 
realm. 

From  these  two  examples  it  is  easily  seen  that  3  and  1  — V —  5 
are  also  primes  in  k(  V — 5). 

§  7.  Failure  of  the  Unique  Factorization  Theorem  in  k  ( V —  5)  • 
Introduction  of  the  Ideal. 

We  shall  now  attempt  to  establish  the  unique  factorization 
theorem  for  the  integers  of  k  ( V —  5 )  and  begin  as  in  the  fore- 
going realms  by  endeavoring  to  prove  the  following  theorem : 

Theorem  A.  //  a  be  any  integer  of  &(V — 5),  and  f3  any 
integer  of  &(y — 5)  different  from  o,  there  exists  an  integer  p 
of  k(  V — 5)  such  that 

n[a  —  (jLp]<n[l3]. 

Let  .    a/p  =  a  +  by~=5, 

where  a  =  r  -\-rx,     b  =  s  +  slt 

r  and  s  being  the  rational  integers  nearest  to  a  and  b,  respectively, 
and  hence 

Let  fl  =  r-{-s\/~^5; 

then  a/p  —  fJL  =  r1-\-s1  ^— ^5 , 

whence  n[a//3  —  fi]  =  rx2  -f-  5^i2  i  % 

that  is,  when  fi  is  determined  as  above,  we  may  have  in  &(V — 5) 

n[a/fi  —  fx\  >  1  instead  of  <  1 
as  has  been  the  case  in  the  three  previous  realms.     Hence  the 
integer  fx  chosen  as  above  will  not  necessarily  satisfy  the  require- 
ments of  the  theorem.     The  method  which  has  hitherto  served  us 
for  the  proof  of  this  theorem  therefore  fails. 


248  THE   REALM    k  (  V 5). 

That  this  theorem  actually  does  fail  for  some  integers  of 
fc(V — 5)  is  evident  from  the  following  example. 

Let  a  =  3  and  fi  =  I  +y/^S, 

then 

We  are  to  find  an  integer  /x=,  ^r  +  ^V — 5>  sucn  that 
«[a//3-M]  =  (i-^)2  +  5(-i-y)2<i, 
but  this  is  impossible,  for  it  is  evident  that  for  all  rational  integral 
values  of  y,  including  o,  the  term  5( — ^  —  y)2  is  itself   >  i. 
The  method  of  proof  adopted  for  Theorem  A  is  seen  to  be  depen- 
dent upon  the  general  form  of  the  norm  of  a  number  rx  -f-  s^o, 
where   I,  w  is  a  basis  of  the  realm.     We  have  thus  in  k(i), 
&(V  —  3),  k(y/2)  and  &(V  —  5)  respectively 
\n[r1  +  s1<»]\  =  \r12  +  s12\,    \rf—  r^+sfl    \rt2  —  2SX%   and 

\rt2  +  SsS\> 

and  the  method  is  successful  if 

KU4>    ft! si 

be  a  sufficient  condition  for 

Wft+A«ii  <  h 

which  is  seen  to  be  the  case  in  k(i),  &(V — 3)  and  &(\/2)  but 
not  in  &(V — 5). 

We  can  easily  determine  all  quadratic  realms  in  which  this 
method  of  proof  holds ;  that  is,  those  in  which  this  way  of  select- 
ing 11  is  always  successful. 

Let  k(-\Jm)  be  any  quadratic  realm,1  v 'm  being  a  root  of  the 
equation  x2  —  m  —  0,  where  m  is  a  positive  or  negative  rational 
integer  containing  no  squared  factor. 

When  m  =  2  or  3,  mod  4,  k(^m)  has  as  a  basis  1,  \/m,  and 
when  m=  i,mod  4,  k(\^m)  has  as  a  basis  1,  ( —  1  +  y/ni)/2  (see 
chap.  X,  §6). 

In  the  first  case,  it  is  easily  seen  that 

1  See  Chap.  X,  §  1. 


THE    REALM    k(\/ 5).  249 


and  in  the  second, 

a/£  —  n=r1  +  s1(—  1  +  \/m)/2, 
which  give  respectively 

n  [  rt  -\-  sx  V  m  ]  =r12  —  m sx 2 


1  1 


and 


»     r1-\-s1 —      l=*t—Vi T~s* 


Considering  first  the  case  m  =  2  or  3,  mod  4,  we  see  that 

is  a  sufficient  condition  that 

\rx2  —  msx2\  <  1  2) 

when  m= — 1,  2,  — 2  or  3;  but  when  \m\  >  3,  then  1)  is  evi- 
dently not  a  sufficient  condition  for  2).  Considering  now  me  I, 
mod  4,  we  see  that  1 )  is  a  sufficient  condition  that 


m  —  1 
-^1  — 


—  sx-\  <  1 


4 
when  and  only  when  m  =  —  3,  5  or  13. 

Hence  Th.  A  and  consequently  the  unique  factorization  theorem 
holds  in  the  realms  k(i),  &(V — 2),  k(yj2),  &(V3)>  £(V — 3>> 
£(V5)>  ^(V^)-  To  these  can  be  added  &(V — 7)>  f°r  when 
M  =  —  7,  which  is  =  1,  mod  4,  if  to  1 )  we  add  the  condition  that, 
when  simultaneously 

1^1=1  and  \sx\  =1, 
then  the  signs  of  rt  and  s%  are  to  be  chosen  alike,  we  see  that  in 
all  cases 

ki2  —  r1s1  +  2s1-\  <i. 

Hence  the  theorem  holds  for  &(V — 7). 

A  further  slight  modification  in  the  method  of  selecting  /x  will 
enable  us  to  show  that  the  theorem  holds  for  k{  V —  n)- 

It  is  easily  seen  that,  if 

I'll  <  i/V5>     kil  <  I/V5; 
then  \r2  —  r^  +  3^1  <  1.  3) 

Moreover,  if  either  \rt\  or  {s^  or  both  =  \,  then  we  can  choose 
the  signs  of  tt  and  sx  so  that  they  are  alike,  and  hence  3)  holds. 


2  50  THE   REALM    &(V 5). 

There  remains  the  case 

1/V54  fa  I  <  */*«       I/V5  i  kll  <  :/2  * 

i.  If  rx  and  st  have  like  signs  3)  evidently  holds. 

ii.  If  rt  and  st  have  opposite  signs,  for  r±  we  can  put  r2 
=^+1  or  rx — 1,  according  as  rx  is  negative  or  positive,  hav- 
ing then 


VS 
and  r2  of  the  same  sign  as  s19  in  which  case 

^22-Vi  +  3^i2<  1. 

Hence  Th.  A  holds  for  &(y^Ti). 

It  can  be  easily  seen  that  the  original  method  of  selection,  even 
when  modified  as  above,  will  give  a  suitable  value  of  /x  in  no 
imaginary  quadratic  realms  other  than  those  enumerated  above, 
and  it  is  furthermore  evident  that  these  are  the  only  imaginary 
quadratic  realms  in  which  the  theorem  holds. 

It  will  be  observed,  as  has  been  said  in  k{i),  that  Th.  A  is 
equivalent  to  saying  that  in  a  given  realm  we  can  find  for  any 
integer  /?  a  complete  residue  system  such  that  the  norms  of  all 
the  integers  composing  it  are  less  in  absolute  value  than  n[(3]. 
This  point  of  view  is  illustrated  graphically  in  Chap.  V,  §  8. 
It  must  be  carefully  noticed,  however,  that  although  Th.  A  is  a 
sufficient  condition  for  the  validity  of  the  unique  factorization 
theorem,  it  is  not  a  necessary  condition,  as  will  be  shown  later. 
The  proof  of  the  theorem : 

Theorem  B.  If  a  and  (3  be  any  two  integers  of  &(V — 5)> 
prime  to  each  other,  there  exist  two  integers,  £  and  q,  of  &(V — 5) 
such  that 

has  been  heretofore  based  upon  Theorem  A,  which  has  been  seen 
not  to  hold  for  &(V — 5).  This,  however,  would  not,  of  course, 
justify  the  assumption  that  Th.  B  does  not  hold  for  &(V — 5), 
Th.  A  being  a  sufficient,  but,  as  we  shall  see  later,  not  a  necessary, 
condition  for  the  validity  of  Th.  B.     Nevertheless,  the  following 


THE    REALM    k(\/ 5).  25  I 

simple  example  will  show  that  Th.  B  does  not  hold  in  general  for 
the  integers  of  &(V — 5). 

Let  a  =  3,     /?=i+V=r5- 

We  have  already  seen  (§6)  that  3  and  1  -j-V — 5  are  prime 
numbers ;  moreover,  they  are  not  associates.  Therefore  they  are 
prime  to  each  other.  We  shall  show  that  it  is  impossible  to  select 
two  integers,  $,  =x-\-y\/ — 5,  and  77,  =f*-|-yy — 5,  such  that 

a$  +  pr,=  i  4) 


H       3(*  +  3'V—  5)  +  (i+V—  5)(«  +  W— 5)  =  i, 
then  3.1-  -\-  a  —  $v=lt 

and  hence  3-r  —  33'  —  Ov  =  1 , 

which  is  impossible  since  the  first  member  only  is  divisible  by  3. 
Therefore  £  and  rj  can  not  be  found  so  as  to  satisfy  4)  and  the 
theorem  does  not  in  general  hold  for  the  integers  of  k{  V — 5). 

We  shall  see  later  (p.  316)  that  the  theorem: 

Theorem  C.  //  the  product  of  two  integers,  a  and  (3  of 
fc(V — 5)  be  divisible  by  a  prime  number,  w,  at  least  one  of  the 
integers  is  divisible  by  r,  which  is  a  necessary  as  well  as  sufficient 
condition  for  the  unique  factorization  theorem,  requires  Th.  B 
as  a  necessary  condition  for  its  validity.  The  following  example 
will  suffice  to  show  that  Th.  C  and  the  unique  factorization 
theorem  do  not  hold  for  the  integers  of  &(V — 5).     We  have 

6  =  2-3=(i+V:^5)(i-V-l), 
and  we  have  shown  (§6)  that  2,  3,  1  +V — 5  and  1  — V — 5  are 
prime  numbers  in  &(V — 5).  Moreover,  the  factors  of  one 
product  are  not  associated  with  the  factors  of  the  other.  There- 
fore 6  is  represented  in  tzvo  zvays  as  the  product  of  prime  factors. 
That  this  is  not  merely  a  peculiarity  of  6  is  seen  from 

21=3.7  =(i+2V:zr5)(i  —  2V^5), 
9=    32=(2+  V=5)(2—   V^5), 
and  49=    72=(2_|_3y_^)  (2  —  3^/^5), 

the  factors  in  the  above  products  being  easily  proved  to  be 
primes  of  &(y — 5). 


2  52  THE   REALM    fc(V 5)- 

Moreover,  that  this  failure  of  the  unique  factorization  law  does 
not  occur  in  &(V — 5)  alone  may  be  shown  by  an  examination 
of  the  realms  k(\/~— -23)  and  fe(V — 89),  in  which  we  have 
respectively 

27  =  38  =  (2 +V—  23)(2—  V—  23), 


and  i25  =  53=(6+V-89)(6-V— 89): 


3,2+  V —  23  and  2  — V —  23  being  prime  numbers  of  fc  ( V —  23  )  > 
and  5,  6+V— -89  and  6 — V — 89  being  prime  numbers  of 
fc(V-89). 

It  can  now  be  made  clear  why  we  could  not  define  the  greatest  com- 
mon divisor  of  two  integers,  a  and  /3, 

i.  As  the  common  divisor,  8,  of  greatest  norm. 

ii.  As  the  common  divisor,  8,  such  that  a/5  and  j8/S  are  prime  to  each 
other. 

If  a=r  (1 -y-^)  (1 +y=5)2  =  6(1 +V-D.  and /3  =  2(1-^/^5), 
then  the  common  divisors  of  a  and  j8  other  than  the  units  are  2  and 
1  —  y/  —  5.  Of  these  1  —  yj  —  5  has  the  greater  norm,  6,  but  1  —  y/  — 5 
is  not  divisible  by  2.  Hence  8  so  determined  has  not  the  important 
property  of  being  divisible  by  every  common  divisor  of  the  two  integers. 

Considering  the  definition  ii  we  see  that  there  are  two  values  of  8,  2 
and  1 — V — 5>  which  satisfy  it,  for  a/2  and  P/2  are  prime  to  each  other, 

and and ■•  ■  ■■   have  the  same  property.     Hence  the  defini- 

1— V— 5  1— V— 5 

tion  ii,  in  addition  to  not  determining  8  so  that  it  is  divisible  by  every 
common  divisor  of  a  and  ft,  does  not  even  determine  it  uniquely.  It  is 
interesting  to  see,  however,  that,  if  we  can  find  in  any  realm  a  common 
divisor,  8,  of  two  integers  a  and  P,  such  that  every  common  divisor  of 
a  and  /3  divides  8,  then  8  will  satisfy  both  the  requirements  i  and  ii ;  for, 
considering  i,  if  8X  be  a  common  divisor  of  a  and  /3  it  divides  8;  that  is, 

8  =  «!/*, 

whence  w[8]  =  rajA]  .  w[>], 

and  therefore  either  |w[8i]  |  <  |n[3]  | 

or  |»[»i]  I  =  !»[«]  I- 

In  the  latter  case 

n|>]  =  ±  1, 

and  hence  a*  is  a  unit ;  that  is  8  and  82  are  associated.    Hence  8  satisfies  i. 
Considering  ii,  we  have 

a  =  8<3i      and      §  =  8/3^ 
Now  if  at  and  A  be  not  prime  but  have  a  common  divisor,  61}  then  8  would 


THE   REALM    &(\/ 5).  253 

not  be  divisible  by  every  common  divisor  of  a  and  j3,  for  it  would  not  be 
divisible  by  Sdt. 

We  now  ask  whether  it  would  be  possible  to  deduce  for  the 
integers  of  &(V — 5),  without  the  use  of  the  unique  factorization 
theorem,  the  series  of  theorems  which  have  flowed  from  it  for 
the  integers  of  R,  k(i),  &(y — 3)  and  &(\/2). 

It  is  easily  seen  that  in  general  these  theorems  do  not  hold  in 
k  (  V —  5 ) .  For  example,  the  analogue  for  k  (  V —  5  )  of  Fermat's 
theorem  would  be: 

//  7r  be  any  prime  of  k(  V — 5)  and  a  any  integer  not  divisible 
by  7r,  then 

a»[7T]-l I  EE=  O,     mod    7T, 

and  indeed,  if  .  v 

tt  =  2  and  a=i-\-2y/ — 5, 
2  being  a  prime  and  1  +  2  V —  5  evidently  not  divisible  by  2,  we 
have 

(i  +  2V:ir5)n[2]-1  — i  =  (r  +  2V— 5)3  — 1 

=  —  60  —  34V— \5  =  o,  mod  2; 
that  is,' the  theorem  holds  in  this  case  . 

But  if  7T  =  2  and  a=i+V — 5> 

we  see  that,  although  2  and  1  +V — 5  satisfy  the  requirements 
2  a  prime  and  1  -f  V — 5  not  divisible  by  2, 
(1  +  V  —  5)"™-1  —  1  =  (1  +  V-^5)3—  1 

=  — 15— 2V— l^o,  mod  2. 

The  cause  of  this  peculiar  difference  in  the  behavior  of 
i+2V — 5  and  1  +V — 5  towards  2  in  this  relation  will  be 
made  clear  later  (p.  379).  Our  next  thought  is  can  we  by  the 
introduction  of  a  new  conception  of  numbers  reestablish  the 
unique  factorization  law  for  the  integers  of  fc(V — 5)  when  the 
factorization  is  expressed  in  terms  of  these  new  numbers.  The 
introduction  of  the  so-called  ideal1  numbers  accomplish  this,  the 
primes  of  fc(V — 5)  being  in  this  widened  number  domain  no 
longer  in  general  looked  upon  as  primes,  but  as  being  factorable 

1  The  term  ideal  number  is  used  here  in  a  general  sense  and  is  not  to  be 
taken  to  refer  particularly  to  the  ideal  numbers  of  KTImmer. 


254  THE   REALM    &(V 5). 

in  terms  of  these  ideal  numbers.  When  this  factorization  has 
been  performed  we  shall  find  that  every  integer  of  fc(V — 5) 
can  be  represented  in  one  and  only  one  way  as  the  product  of 
prime  ideal  numbers. 

The  following  considerations  will  make  clearer  their  nature, 
and  the  ideas  which  have  led  to  their  conception.  Let  us  con- 
sider the  narrowed  number  domain  composed  of  all  positive 
rational  integers  congruent  to  1,  mod  5;  that  is, 

1,  6,  11,  16,  21,  26,  31,  36,  41,  46,  etc.  5) 

Our  definitions  of  divisibility  and  prime  number  being  the  same 
as  before,  we  see  that,  when  our  operations  are  confined  to  num- 
bers of  this  domain,  the  unique  factorization  law  does  not  in 
general  hold ;  for  example, 

336=   6-56  =16. 21, 

1806  =  21-86  =   6-301, 

1296=  64=i6-8i, 
and  6,  16, 21,  56,  81, 86  and  301  are  easily  seen  by  multiplication  of 
the  numbers  5)  to  be  prime  in  this  domain.  The  cause  of  this 
failure  of  the  unique  factorization  law  is  at  once  seen  to  lie  in 
the  absence  of  the  remaining  positive  integers.  As  we  suppose 
these  integers  to  be  unknown  to  us  and  in  fact  to  have  no  real 
existence,  we  ask  by  what  train  of  reasoning  are  we  led  from  the 
requirements  of  the  task  to  be  accomplished,  that  is,  the  reestab- 
lishment  of  the  unique  factorization  law,  to  the  introduction  of 
these  missing  integers,  or  rather  the  introduction  of  symbols 
which  have  their  properties  so  far  as  the  task  in  hand  is  concerned. 
Consider  336  =  6-56=16-21. 

Since  6  is  not  contained  in  either  16  or  21,  although  the  product 
16-21  is  divisible  by  6,  we  suppose  6  to  be  the  product  of  two 
factors  one  of.  which  is  contained  in  16,  the  other  in  21,  and 
denote  these  factors  by  (6,  16)  and  (6,  21),  respectively.  The 
factor  (6,  16)  plays  the  same  role  with  respect  to  6  and  16  in 
all  questions  of  divisibility  in  which  these  new  numbers  are  used 
that  the  greatest  common  divisor  of  two  integers  plays  with  re- 


THE   REALM    &(V 5).  255 

spect  to  these  integers  when  only  the  original  numbers  of  the 
domain  are  involved.  We  can  therefore  in  this  sense  consider 
(6,  16)  as  the  greatest  common  divisor  of  6  and  16.  Likewise 
we  consider  (6,  21)  as  the  greatest  common  divisor  of  6  and  21, 

and  we  write 

6=(6,i6)(6,2i), 

denoting  by  this  equation  that  6  and  the  product  (6,  16)  (6,  21) 
in  all  questions  of  divisibility  play  the  same  role;  that  is,  every 
integer  that  is  divisible  by  6  is  divisible  by  (6,  16)  (6,  21),  and 
conversely.  This  convention  is  evidently  justified  by  the  fact  that 
in  reality  (6,  16)  is  2  and  (6,  21)  is  3.     Similarly  we  have 

56=  (56,  16)  (56,  21), 

16  =  (16,    6)  (16,  56), 

2I  =  (2I,-   6)  (21,  56), 

and  hence 

336  =  6.56=  (6,  16)  (6,  21)  (56,  16)  (56,  21) 
=  16.21  =  (16,  6)  (16,  56)  (21,  6)  (21,  56), 
and  the  factorization  is  seen  to  be  the  same,  the  change  of  order 
of  the  numbers  in  the  parenthesis  having  no  effect  on  the  symbol; 
that  is,  (6, 16)  =  (16, 6),  etc. 

We  have  now  seen  that  the  failure  of  the  unique  factorization 
law  in  a  certain  number  domain  can  be  remedied  by  the  introduc- 
tion of  a  new  kind  of  number  each  of  which  is  defined  by  a  pair 
of  integers  of  the  domain  and  may  be  looked  upon  as  the  greatest 
common  divisor  of  these  integers.  These  numbers  might  be 
called  the  ideal  numbers  of  the  domain,  and  although  the  fact 
that  the  numbers  of  this  domain  do  not  form  a  realm  prevents 
our  expanding  their  conception  and  definition  to  the  extent  that 
we  shall  now  develop  those  of  the  ideal  numbers  of  &(V — 5), 
still  we  shall  find  that  the  same  conception  will  enable  us  to 
reestablish  the  unique  factorization  law  in  this  realm.  We  shall 
not,  however,  conceive  of  these  new  numbers,  which  we  are  about 
to  introduce  into  k(\/ — 5),  simply  as  being  each  the  greatest 
common  divisor  of  a  pair  of  integers  of  k{  V — 5)  and  as  defined 
by  these  integers,  but  as  being  each  the  greatest  common  divisor 


256  THE   REALM    k(\/ 5). 

of  an  infinite  system  of  integers  of  &(V — 5)  and  as  defined  by 
any  finite  number  of  these  integers  such  that  all  other  integers  of 
the  system  are  linear  combinations  of  these  with  coefficients 
which  are  any  integers  of  the  realm.  These  numbers  we  shall 
call  the  ideal  numbers,  or  briefly  the  ideals  of  &(V — 5).  To 
make  this  clearer,  consider  the  equation 

2-3=(i+vzr5;)(i—  v^s). 

Since  2  divides  neither  (1  +V — 5)  nor  (1  — V — 5),  although 
it  divides  their  product,  we  must,  to  reestablish  the  unique  factori- 
zation law,  consider  2  as  the  product  of  two  ideal  factors,  a  and 
h,1  which  divide  1  +V — 5  and  1 — V — 5  respectively,  the  quo- 
tients being  supposed,  of  course,  to  be  ideal  numbers  also.  We 
can  denote  a  and  h  by  the  symbols  (2,  1 +V — 5)  and  (2» 
1 — V5)  respectively.  If  now  a  be  considered  to  bear  the  rela- 
tion of  greatest  common  divisor  to  2  and  1  -f-V — 5>  it  will  bear 
this  relation  to  the  entire  system  of  integers,  which  are  linear 
combinations  of  2  and  1 -f-V—  5;  that  is,  those  of  the  form 
2a -\-  (1  +V — 5)/?,  where  a  and  j3  are  any  integers  of  the  realm. 
Conversely,  if  a  be  considered  to  bear  this  relation  to  the  entire 
system,  it  will  bear  it  to  2  and  1  +V — 5-  We  consider  then  a 
to  be  determined  not  by  2  and  1  -f-V — 5  alone  but  by  this  entire 
system  of  integers,  and  by  a  natural  transition  say  now  that  a 
is  this  system  of  integers. 

We  write  therefore 

understanding  by  this  symbol  the  entire  system  of  integers  which 
are  linear  combinations  of  2  and  1 +V — 5>  w^h  coefficients 
which  are  any  integers  of  the  realm.  In  order  to  define  a,  it  is 
therefore  sufficient  to  give  any  set  of  integers  such  that  all  linear 
combinations,  with  coefficients  as  above,  exactly  constitute  the 
above  system.  Hence  we  can  introduce  into  the  symbol  defining 
a  any  integer  that  is  a  linear  combination  of  those  already  there, 
and  can  omit  any  integer  that  is  a  linear  combination  of  those 
remaining ;  thus : 
1  Ideals  will  be  denoted  by  German  letters. 


THE   REALM    &(\/ 5).  257 

a=(2,  i+v— 1) 
=  (2,  i+V— "5,  2  +  2^— ~5,  3  +  3V^5) 
=  {2,  2  +  2V=75,  3  +  3V— "5)- 

The  object  of  the  preceding  discussion,  that  has  been  by  no 
means  rigorous,  has  been  first  to  show  the  necessity  for  the  intro- 
duction of  ideal  numbers,  and  second  to  acquaint  the  reader  in 
some  degree  with  the  ideas  which  have  led  to  their  conception  and 
which  induce  us  to  adopt  the  definition  which  we  shall  now  give. 
The  justification  of  this  definition  will  be  found  in  the  fact  that, 
after  we  have  defined  what  is  meant  by  the  equality  of  two  ideals 
and  what  is  meant  by  their  product,  we  shall  see  that,  when  the 
integers  of  &(V — 5)  are  resolved  into  their  ideal  factors,  the 
unique  factorization  law  will  be  once  more  found  to  hold.  More- 
over, we  shall  see  that  the  behavior  of  an  ideal  towards  the  integers 
of  the  system  constituting  it  is  such  as  to  warrant  our  original 
conception  of  an  ideal  as  the  greatest  common  divisor  of  this 
system. 

§  8.    Definition  of  an  Ideal  of  jfe(y— 5). 

An  ideal  of  k(y — 5)  is  an  infinite  system  of  integers  composed 
of  all  linear  combinations  of  any  finite  number  of  integers, 
ax,a2,  '-,an,  the  coefficients  being  any  integers  of  the  realm.1 

The  integers  ax,a2,  --.an  are  said  to  define  the  ideal  and  the 
integers  of  the  infinite  system  of  integers  constituting  the  ideal 
are  called  the  numbers  of  the  ideal.  If  an  ideal  a  be  defined  bv 
the  integers  a^a^,  •••,<*»  we  write 

a=  (alta2,  •••,a»), 
understanding  thereby  the  infinite  system  of  integers  of  the  form 

SA+$AH h&flt*  1) 

where  £x,  £2,  ••-,£«  are  any  integers  of  the  realm.     We  shall  call 
(alf  a2,  •  •  •,  a„)  the  symbol  of  the  ideal  of  a. 

^^The  general  definition  of  an  ideal  of  any  quadratic  realm  (Chap.  XII, 
§1)  seems  at  first  sight  broader  than  this  definition,  but  as  it  is  shown 
that  all  the  numbers  of  any  ideal  are  linear  combinations  of  a  finite  num- 
ber of  them,  the  definitions  are  equivalent. 

17 


258  THE   REALM    fe(V 5)- 

If  y  be  one  of  the  integers  included  in  i)  ;  that  is,  if 

y  ==  kfit  +  X2a2  -f-  •  •  •  -L.  XnCLn, 

where  A^  A2,  •  •  •,  Xn  are  integers  of  the  realm,  we  have 

a=(a1,a2,  •  •  •, a»)  =  (alf a2,  •■••,a»,y),  2) 

for  the  infinite  system  of  integers  of  the  form 

lyA'H-  ^2a2  + h  ynan  +  ^w+1y,  3) 

where  t)x,-q2,  ■•■,r}n+1  are  any  integers  of  the  realm,  is  the  same  as 
the  system  1),  since  putting  the  value  of  y  in  3),  we  have 

ill  +  Vn+iK)^i  +  (V2  +  yn+1X2)a2  -j h(Vn  +  r)n+lXn)an, 

a  system  that  evidently  coincides  with  1).  It  is  evident  then 
from  2)  that  we  may,  without  changing  an  ideal,  introduce  into 
its  symbol  any  integer  which  is  a  linear  combination  of  those 
already  there,  the  coefficients  being  integers  of  the  realm,  and 
may  omit  from  the  symbol  any  integer  which  is  a  linear  combi- 
nation of  those  remaining. 

§  9.    Equality  of  Ideals. 

Two  ideals,  a=  (alt  a2,  •  -,am)  and  B  =  (filf  yG2,  •  •  •, £»),  are 
equal  when  the  two  infinite  systems  of  integers  that  constitute 
these  ideals  are  the  same.  The  necessary  and  sufficient  condition 
for  this  is  that  every  number,  ax,a2,  •■•,am,  defining  a  shall  be 
linear  in  the  numbers,1  ft,/?2,  •••,£»»,  defining  B,  and  that  every  /? 
shall  be  linear  in  the  a's ;  that  is,  it  is  necessary  and  sufficient  that 
we  shall  be  able  to  introduce  the  numbers  ax,a2,  •••,a«  into  the 
symbol  of  B,  and  the  numbers  /3X,  (32,  '-,(3n  into  the  symbol  of  a; 
in  other  words,  zve  must  be  able  to  reduce  the  symbol  of  either 
one  of  the  ideals  to  that  of  the  other. 

Ex.  1.  To  prove  that  the  two  ideals  a  =  (2,  1  +  V — 5),  and 
h=  (2,  1  —  V — 5)>  are  equal.    We  have 

(2,  i  +  V:i:5)  =  (2,  i  +  V1^   1  — V— ~5), 

since  1  —  v^  =  2(—  Vj^S)  +  (1  +  V-HS)  I 

and  (2,  1  +  y/^5,  1  —  y/^l)  =  (2,  1  —  V^S)  5 

since  I  +  V^-S  —  (y/—$2^  (t  —  y/ZZJ). 

1  When  we  say  that  cu  is  linear  in  ft,  ft,  •••,  /Sw  we  shall  understand  that 
a%  =  lift  -f  |2ft  H h^»>  where  &,  l2,  ■••,  In  are  integers  of  the  realm. 


THE    REALM    fc(V 5).  259 

Having  reduced  the  symbol  of  a  to  that  of  B,  the  two  ideals  are  seen 
to  be  the  same. 

Ex.  2.  To  prove  that  the  two  ideals  a=  (3,  1 +V — 5),  and  h=  (3, 
1 — V — 5),  are  unequal. 

If  we  can  show  that  any  number,  as  1  -f-V —  5,  of  a  is  not  a  number 
of  b,  the  two  ideals  will  evidently  be  unequal.  If  1  -f-  V —  5  be  a  number 
of  B,  then  two  integers,  x  +  yV — 5,  u-\-v\J — 5,  of  &(V — 5)  must  exist 
such  that 

1  +  V^S  =  U  +  y  V  — 1)3  +  O  +  vy/^S)  (1  —  V  — 5), 

and  hence  1  =  3*  -f-  «  +  5^, 

1  =  3y  +  »  —  u, 

whence  by  addition  2  =  $x  -f-  3y  -f-  6v, 

an  equation  between  rational  integers  that  is  impossible,  since  3  is  a 
divisor  of  the  right  hand  member  but  not  of  the  left  hand  member. 

Hence  the  required  integers  do  not  exist,  and  1  +  V —  5  is  therefore 
not  a  number  of  the  ideal  b.     The  ideals  are  therefore  unequal. 

Ex.  3.  To  prove  that  the  two  ideals  a  =(2,  i  +  V — 5)>  and 
b=(4,  2-\-2\J — 5),  are  unequal. 

Although,  as  is  easily  seen,  the  numbers  denning  the  second  ideal  may 
be  introduced  into  the  symbol  of  the  first  ideal,  we  cannot  introduce  the 
number  2  of  the  first  ideal  into  the  symbol  of  the  second ;  that  is,  we  can- 
not find  two  integers,  x  +  yyj —  5,  u  -f  v\J —  5,  such  that 

2  =  O  +  y  VZr5)4  +  («  -f  v^/=5)  (2  +  2yJ~—$), 
for  from  this  equation  it  would  follow  that 

2  =  4X  +  2M  —  IOZ/, 

o  ==  4y  -f-  2M  -f-  2V, 

whence  by  subtraction         2  =  4-r  —  4y  —  \2v, 

an  equation  in  rational  integers  that  is  impossible,  since  4  is  a  divisor  of 
the  second  member  but  not  of  the  first  member.     The  two   ideals  are 
therefore  unequal. 
Ex.  4.     Show  that 

(2,  1  +V=5)  =j=  (3,  1  +V^5). 
Ex.  5.     Show  that 

(29,  32  — 27Vz:r5)  =  (3  +  2V=r5)- 
Ex.  6.     Show  that 

(49,  21  —  /V— 5,  2I  +  7^J^rS,  14)  ==  (7). 
Ex.  7.     Show  that 

(3  —  V^l.  1  +  2y/^J)  =  {7,  3  —  V:Zr5)- 


260  THE   REALM    k(\/ 5). 

§  10.    Principal  and  Non-Principal  Ideals. 

If  among  the  numbers  of  an  ideal,  a,  there  exist  a  number,  a, 
such  that  all  numbers  of  the  ideal  are  multiples  of  a,  then  a  is 
said  to  be  a  principal  ideal,  and  we  have 

o=(a). 

If  such  a  number  does  not  exist,  a  is  said  to  be  a  non-principal 
ideal.  The  necessary  and  sufficient  condition  for  a  to  be  a  prin- 
cipal ideal  is  evidently  that  we  shall  be  able  to  introduce  into  the 
symbol  of  a  a  number  a  such  that  all  the  numbers  defining  a  are 
multiples  of  a.  If  such  a  number  cannot  be  introduced,  a  is  a 
non-principal  ideal.  Let  us  consider  a  few  ideals  with  a  view  to 
determining  whether  they  are  principal  or  non-principal  ideals. 

i.   (7)^(2 +  V^5),     (6,  8,  2  +  6  V=S),     (3,  3V=S), 
(3,  V— 5),  (5,  V— 5)- 
ii.  (2,  i+V— 5),  (3>  i+V—  5),  (3,  1—  V— _5)- 
Considering  those  of  the  set  i,  (7)  and  (2 +V — 5)  are  seen 
at  once  from  the  definition  to  be  principal  ideals ;  also 

(6,  8,  2  +  6V— 1)  =  (6,  8,  2  +  6V— 1,2)  =  (2), 

(3,  3V— ~5)  =  (3), 
(3,  Vzr5)=5:(3i  V— "5,— 5)  =  (3J  VF-S>.— 5-  0=<l)> 

(5,  V-1)  =  (V-1). 

Hence  all  ideals  of  the  first  set  are  principal  ideals. 

Consider  now  the  ideals  of  the  set  ii.  If  (2,  1 +V — 5)  be 
a  principal  ideal,  then  there  must  exist  a  number,  a,  of  the  ideal 
such  that  2  and  1  +V — 5  are  both  multiples  of  a. 

The  numbers  2  and  i"+V — 5>  being  primes  in  &(V — 5)  and 
not  associated,  have  as  their  only  common  divisors  zb  I.  Hence  a 
must  be  1  or  —  1. 

Since,  if  1  be  a  number  of  the  ideal,  —  1  is  also  one  of  its 
numbers  and  vice  versa,  it  is  sufficient  to  see  whether  we  can  find 
two  integers  x  +  y  V —  5  and  u  +  ^ V —  5,  such  that 

1  =  2<>  +  3'V— 5)  +  (1  +V—  5)  O  +  ^V—  5).         0 
We  have  from  1 )        1.  =  2x  +  u  —  5V> 
o  =  23/  -j-  v  +  u, 


THE   REALM    &(\/ 5).  26  I 

which  give  by  subtraction 

I  sa  2X  —  23/  —  6v, 

an  equation  in  rational  integers  that  is  impossible,  since  the  second 
number  only  is  divisible  by  2.  Hence  1  is  not  a  number  of  the 
ideal  (2,  1  -|~V — 5),  and  this  ideal  is  therefore  a  non-principal 
ideal. 

Ex.  1.  Show  in  like  manner  that  (3,  i  +  V — 5)  and  (3»  J  —  V — 5) 
are  non-principal  ideals. 

Ex.2.  Show  that  (7,  i  +  2n/—"S)  and  (7,  1  —  2y/ — 5)  are  non- 
principal  ideals. 

Ex.  3.     Show  that  (21,  9  +  3 V — 5,  — 2  +  4V — 5)  is  a  principal  ideal. 

Had  we  introduced  the  conception  of  the  ideal  in  the  realms 
k(i),  k(  V — 3)  and  k(\/2),  we  should  have  seen  that  in  all  these 
realms  every  ideal  is  a  principal  ideal,  for  if  a,  =  (alfa2t  •••,a„), 
be  an  ideal,  defined  as  above,  of  any  one  of  these  realms,  then, 
since  the  unique  factorization  law  holds  in  all  these  realms,  we 
could  in  every  case  find  integers  £r,€2,  -',£n  such  that 
&0,  +  i2a2  -\ f-  inPtn  =  5, 

where  8  is  the  greatest  common  divisor  of  ax,a2,  "',On*  Hence 
we  have  a=  (alt a2,  ••-,an,8)  =  (8), 

a  principal  ideal. 

On  the  other  hand,  we  have  seen  (Th.  B)  that  it  is  not  always 
possible  in  &(V — 5)  to  find  the  integers  £lf  £2,  ••-,£»;  hence  the 
fact  that  not  all  ideals  of  &(V — 5)  are  principal  ideals. 

§  11.    Multiplication  of  Ideals. 

By  the  product  of  two  ideals 

a,=  (alta2f  •••,aw),  and  6,=  (&,&,  •••,£«), 
we  understand  the  ideal  defined  by  all  possible  products  of  a  num- 
ber defining  a  by  a  number  defining  h;  that  is, 

ab  =  (ajlv axp2,  •  "lOifin,  - -'tamplf  •  •  •, aW?„). 

In  other  words,  the  product  of  a  and  b  is  the  ideal  whose  numbers 
are  all  possible  products  of  a  number  of  a  by  a  number  of  B, 
together  with  all  linear  combinations  of  these  products.  It  is  evi- 
dent from  the  above  definition  that  the  commutative  and  asso- 


262  THE   REALM    k(\/ 5). 

ciated  laws  hold  in  the  multiplication  of  ideals;  that  is,  ah  =  ha 
and  ab-c  =  a-hc. 
Ex.  1. 

(3,  1  + V—5)(3,  1  —  V—  5)  =  (9,  3  —  3V— 5>  3  +  3\/-r5'  6>- 

=  (9,  3  —  3^—^3  +  3^—5,  6,  3), 
=  (3). 
Ex.  2.         (2,  1  +  V^S)"=  (2,  1  +  ^trj)  (2,  j  +  VZT5), 
=5  (4,  2  +  2^=5,  —4  +  2^/^5), 

=  (4,  2  +  2y— 1,  _4  +  2y^5,  2), 

since  2  +  2\/ —  5  —  ( —  4  +  2\/ —  5)  —  4  =  2.    Hence,  since  all  numbers  in 
the  symbol  are  multiples  of  2,  which  is  a  number  of  the  symbol, 

(2,  i  +  V-5)2=(2). 
Ex.  3. 
(2,  i+y^5)(3,  !+y~5)  — (6,  2  +  2^=5,  3  +  3\/—  5^  — 4+2\/—  5) 

=  (6,  2  +  2^=5,  3  +  3y—  5,  i+y-^), 
since  2  +  2V^r5  —  6  =  —  4  +  2V= T 

and  3  +  3  V^  —  (2  +  2  %/— 5)  =  1  +  V^ 

whence,  since  all  numbers  in  the  symbol  are  multiples  of  1  +  V —  5, 

(2,  1  +  V- 5)  (3,  1  +  V-  5)  =  (I  +  V-  5). 
Ex.  4. 

(2,  i-f-y~5)(3,  1  — V—  5)  =  (6,  2  —  2y—  1,  3  +  3^/^5,  6) 

=  (6,  2  — 2^=5,  3+3-^—5;  i—y—g), 
since  6—  (2  —  2V=r5)  —  (3  +  3Vzr5)  =  I  —  V1^ 

whence,  since 

3  +  3V—  5  =  6  —  (2  —  2  V—  5)  —  ( 1  —  V— 1) . 
(2,  1  +  y— 5)  (3,  1  —  V  —  1)  =  (6,  2  —  2 y  — 1,  1  —  v— "5) 

=  (i-y-5)> 

since  all  the  numbers  in  the  symbol  are  multiples  of  1  —  \J —  5. 
Ex.  5.     Show  that 

a)  (3,  1  +  2V:=r5)  (3,  1  —  2V:zr5)  =5=  (3), 

b)  (7,  1  +  2V-TK7,  I  —  2\/— 5)-(7), 

c)  (3,  i  +  2V-~5)(7,  l+2\fi^$)=(l  +  2y/^J), 

d)  (3,  1  —  2\/zr5)(7,  1  — 2  V— 1)  =  (1  — 2  V— 5). 


THE   REALM    k(\/ 5).  263 

§  12.     Divisibility  of  Ideals. 

An  ideal,  a,  is  said  to  be  divisible  by  an  ideal,  h,  when  there 
exists  an  ideal,  c,  such  that 

a  =  Bc; 
b  and  c  are  then  said  to  be  divisors  or  factors  of  a. 

§  13.     The  Unit  Ideal. 

Every  ideal  a,  =  (ax, a2,  •  •  -,am),  of  &(V — 5)  is  divisible  by 
the  ideal  (1),  for 

ct(  1 )  =  (alt  a2,---,an)  ( 1 )  =  (alf  a2,  •  •  •,  a»)  =  a. 

That  (1)  is  the  only  ideal  of  fc(V — 5)  possessing  this  property 
can  be  easily  shown. 

Suppose  that  there  is  another  ideal  b=  (8X,82,  •••,§„),  which  is 
a  divisor  of  every  ideal  of  fc(V — 5)-  Since  it  divides  the  ideal 
( 1 ) ,  we  must  have  ( I )  =  b'm, 

where  m=  Oi,f4,  •••,**«•)• 

Then  (1)  =  (Sx,  82,  •••,8n)(/*i,/*2»  •"•*f»»)j 

and  hence        1  =  g^ft,  -f  £A/%  H 1-  £m„8„/>im  1 ) 

=  AjSi  -p-  A282  H~~  '  *  *  ~T~  An8n, 

where    ivit,  ■■■,$mn    and    hence    a^a.,,  --jAn    are    integers    of 
&(V — 5).     Therefore  1  is  a  number  of  b  and 

b=(81,82,-..,8n,i)  =  (i). 

The  ideal  (1)  is  therefore  the  only  ideal  which  divides  every 
ideal  of  k  (  V —  5  )  •  Hence  it  is  called  the  unit  ideal  of  &  (  V —  5  )  • 
It  is  evidently  the  whole  system  of  integers  of  fc(V — 5)-  It 
should  be  noticed  that  from  1 )  it  follows  also  that  1  is  a  number 
of  m,  and  in  general  we  may  show  by  this  method  that,  if  an 
ideal  a  be  divisible  by  an  ideal  h  then  all  numbers  of  a  are  num- 
bers of  h. 

§  14.    Prime  Ideals. 

An  ideal  different  from  (1)  and  divisible  only  by  itself  and  (1) 
is  called  a  prime  ideal.  An  ideal  with  divisors  other  than  itself 
and  (1)  is  called  a  composite  ideal. 


264  THE   REALM    &(V 5). 

We  shall  show  that  (2,  1  +V — 5)  is  a  prime  ideal.  If  this 
be  not  the  case,  two  ideals,  a  and  5,  neither  of  which  is  (1),  must 
exist  such  that 


(2,  i+V—  5)  =  aB. 
Let  a  =  (a19  a2,  •  • .,  am),    B  =  (ft,  0*  •  •  •,  0„). 

Then  we  should  have 

(2,  i+V— l)  =  (a15a2,  ...,aw)(^,^2,  ...,/?n). 
It  may  be  shown  now  by  the  method  employed  in  the  last 
paragraph  that  2  and  1 +V — 5  are  numbers  of  each  of  the 
ideals  a  and  b,  and  hence 

(2,  I  +  V— 5)  =  («1»  '  •  'I  Am,  2,  I  +  V—  5) 

(ft,  ...,j8„,2,  i+V—  5). 

Let  at,  =  a  +  &  V —  5,  be  any  one  of  the  integers  alt  a2,  •  •  • ,  am ; 
then  ai  =  Z?(i+V — 5)+° — &• 

But  a  —  b  is  a  rational  integer,  and  hence  is  of  the  form  2c  or 
2c  +  1,  where  c  is  a  rational  integer.     We  have  therefore  either 

ai  =  fc(i+V—  5)  +2^  x) 

or  ai=^fc(I^-yZT5)^2c-f  1.  2) 

If  1 )  be  the  case,  ai  may  be  omitted  from  the  symbol  a.  If  2) 
be  the  case,  we  have 

ai  —  b{i+^~—^s)—  2c=i, 
and  1  may  therefore  be  introduced  into  the  symbol  of  a ;  all  other 
numbers  could  then  be  omitted  and  we  should  have 

a=(i). 
Proceeding  in  this  manner  with  each  of  the  numbers  alt  a2, 
'•',am,  we  see  that  one  of  the  two  following  cases  must  occur, 
either  all  of  the  numbers  axi a2,  '-,am  are  linear  combinations  of 
2  and  1  +V —  5,  and  hence  may  be  omitted  from  the  symool  of  a, 
in  which  case  we  have 

a=(2,  i+V— ~5), 
or  some  number  of   a  is  not  a  linear  combination   of  2  and 
1  +  V —  5>  m  which  case  1  may  be  introduced  into  the  symbol  of 
a  and  we  have 

a=(i). 


THE   REALM    &(V 5)-  265 

The  same  is  evidently  true  for  B.     We  have  therefore  as  the 
only  possible  factorizations  of  (2,  1 +V — 5) 

(2,  i+V=5)  =  (i)(i)  =  (i),  3) 

or  —  (2,  1  +  V^)  (2,  1  +V—  5),  4) 

or  =(2,  i+V=5)(i), 

or  =(i)(2,  i+V— ~5)- 

It  has  already  been  proved  that 

(2,  i+V-5)  +  (i), 
hence  3)  is  impossible. 

Likewise  it  may  easily  be  shown  that  4)  is  impossible,  for  we 
have  seen  (§11)  that 

(2,  i+y=5)==(2), 
while,  since  1  +V — 5  1S  not  a  multiple  of  2, 

(2,  i+v-l)4=(2). 

Hence  4)  is  impossible. 

The  only  divisors  of  (2,  1 +V — 5)  are  therefore  the  ideal 
itself  and  (1).     Hence  (2,  1  +y — 5)  is  a  prime  ideal. 

It  may  be  shown  similarly  that  (3,  1 +V — 5)  and  (3, 
r — V — 5)  are  prime  ideals.  The  proof  in  these  cases  is  sug- 
gested as  an  exercise. 

Ex.  Prove  that  every  ideal  of  the  form  (p,  1  -j-  q\/ — 5),  where  p  and  q 
are  rational  primes  different  from  each  other,  is  a  prime  ideal. 

§  15.  Restoration  of  the  Unique  Factorization  Law  in  Terms 
of  Ideal  Factors. 

We  shall  now  show  that  although  the  factorization  of  6  into 
its  prime  number  factors  in  k{  V —  5)  *s  not  unique,  nevertheless, 
when  we  resolve  the  principal  ideal  (6)  into  its  prime  ideal  fac- 
tors this  factorization  is  unique.1  There  are  evidently  two  differ- 
ent factorizations  of  (6)  into  principal  ideal  factors;  that  is, 

(6)  =  (2)(3)  =  (i+V-5)(i-V^5).  1) 

1  We  speak  of  the  factorization  of  an  integer  a  into  its  ideal  factors, 
meaning  thereby  always  the  factorization  of  the  principal  ideal  (a) 
defined  by  a. 


266  THE   REALM    k(\/ 5). 

These   factors  are,  however,  not  prime  ideals,   for  we  have 
shown  (§11)  that 

(2)  =  {2,  i+V^5)2, 

(3)  =  (3,  i+V-l)(3,  i-V-1), 
(i+V=5)  =  te  r+V^5)(3,  i+V-5), 

and         (1— V—  5)  =  (2,  i+V— 5)(3,  i—  V—  5)- 

We  have  shown  also   (§  14)   that  these  factors  of   (2),   (3), 

(1  +  V — 5)  and  (1 — V — 5)  are  a*l  prime  ideals. 
Substituting  in  1)  we  have 

(6)  =  (2) (3)  =  (2,  i+V-5)2(3.  i+V=5)(3>  1— V^Di 

and 


(6)=(i+V-5)(i-V-5) 


=(2,  i+V-5)(3,  i+V— 5)  (2,  i-V— 5)(3,  1— V— 5) 


=(2,  i+V—  5)2(3,  i+V-5)(3,  1—  V—  5). 

Hence  (<5)  can  be  factored  in  one  and  but  one  way  into  prime 

ideal  factors.  0 

Ex.  Show  that  the  factorizations  of  9,  14,  21,  and  49  into  prime 
number  factors  are  not  unique  but  that  the  factorizations  of  (9),  (14), 
(21),  and  (49)  into  prime  ideal  factors  are  unique. 

We  have  now  shown  that  the  introduction  of  the  conception  of 
the  ideal  in  &(V — 5)  has  accomplished,  at  least  in  the  particular 
example  given,  what  we  desired;  that  is,  the  restoration  of  the 
unique  factorization  law. 

Instead  of  showing  that  the  unique  factorization  law  holds  in 
general  in  &(V — 5)  when  the  factorization  is  expressed  in  terms 
of  prime  ideal  factors,  and  then  investigating  the  properties  of 
the  integers  and  ideals  of  this  realm,  we  shall  proceed  at  once  to 
the  discussion  of  the  general  quadratic  realm  defined  by  the  root 
of  any  irreducible  quadratic  equation.  Among  these  realms  are 
included,  of  course,  the  special  realms  k(i),  &(V — 3),  &(V2) 
and  &(V — 5).  We  shall  see  that  when  the  factorization  in  any 
quadratic  realm  whatever  is  expressed  in  terms  of  prime  ideal 
factors  it  is  unique,  and  we  shall  be  able  to  deduce  general 
theorems  for  the  integers  and  ideals  of  any  realm  similar  to  those 


THE   REALM    £(\/ 5).  267 

found  for  the  integers  of  realms  in  which  the  unique  factorization 
law  held  in  the  ordinary  sense.  We  shall  find,  moreover,  that 
the  introduction  of  the  ideal  will  lead  us  to  the  discovery  of  new 
and  deeper  properties  of  these  realms. 

The  introduction  of  ideal  factors  is  due  to  Kummer,  but  the 
form  used  in  the  text  and  known  as  ideals  is  due  to  Dedekind. 
For  an  account  of  Rummer's  researches  see  his  papers,  Crelle, 
Vol.  XXXV,  pp.  319  and  327,  especially  the  former,  in  which  he 
announces  his  introduction  of  the  ideal  number;  in  the  latter 
paper  he  expands  the  theory.  A  brief  account  of  Rummer's  con- 
ception is  given  in  the  eleventh  supplement  to  Dedekind's  edition  of 
Dirichlet  lectures,  pp.  545-550;  see  also  Bachmann,  Allgemeine 
Arithmetik  der  Zahlenkorper,  pp.  150-160,  for  a  very  interesting 
discussion  of  Kummer's  ideal  numbers  and  other  methods  of 
reinstating  the  unique  factorization  law  in  the  general  algebraic 
number  realm. 


CHAPTER    IX. 
General  Theorems  Concerning  Algebraic  Numbers. 

§  i.    Polynomials  in  a  Single  Variable.1 

Before  beginning  the  study  of  the  general  quadratic  realm  we 
shall  give  a  few  theorems  which  are  necessary  for  our  future 
investigations. 

First  of  all,  we  shall  prove  a  theorem  concerning  the  divisibility 
of  polynomials  in  a  single  variable.  By  a  polynomial  in  a  single 
variable,  x,  is  meant,  as  has  been  said,  an  expression  of  the  form 


a0xn  -f-  a^-1  +  •  •  •  +  a 


n, 


where  n  is  a  positive  rational  integer  and  the  a's  are  quantities 
not  containing  x.  The  sum,  difference  and  product  of  two  poly- 
nomials in  x  are  evidently  polynomials  in  x. 

In  what  follows  we  shall  in  all  cases  assume  the  a's  to  be 
rational  numbers. 

A  polynomial,  f(x),  is  said  to  be  divisible  by  another  poly- 
nomial, /i(-r),  when  a  third  polynomial,  f2{x),  exists  such  that 

/<*)— /«.(*)£(*)• 

It  is  evident  that  all  polynomials  of  the  oth  degree,  that  is,  the 
rational  numbers,  divide  every  polynomial  in  x. 

If  fx{x)  and  f2(x)  have  no  common  divisors  other  than  con- 
stants, they  are  said  to  be  prime  to  each  other,  or  to  have  no 
common  divisor. 

Theorem  i.  //  f1(x)  and  f2(x)  be  two  polynomials  in  x 
without  a  common  divisor,  there  exist  two  polynomials  in  x, 
4>x(x)  and  </>2(-r),  such  that 

1  Weber :  Algebra,  Vol.  I.,  §§  i  to  6. 

268 


GENERAL   THEOREMS   CONCERNING   ALGEBRAIC   NUMBERS.        269 

Let  f±(x)  and  /2(^)  be  of  degrees  m  and  n,  respectively,  and 

m  ^  n. 
By  division  we  may  put  ft  in  the  form 

ft=*qjt+f»  1) 

where  q19  the  quotient,  and  /3,  the  remainder,  are  polynomials  in 
x,  and  /3  is  of  lower  degree  than  /2. 
Likewise  we  may  put  f2  in  the  form 

/2  =  92/3 +/4,  2) 

where  fz  and  f4  are  polynomials  in  x,  and  f4  of  lower  degree 
than  /3. 

Continuing  this  process,  which  is  none  other  than  that  of  finding 
the  greatest  common  divisor  of  fx{x)  and  f2(x),  we  have 

f*=qJ*+U  3) 

U  =  ^^5  +  /«> 

and  arrive  finally  at  a  point  where  the  remainder  is  a  constant, 
/fc,  different  from  o,  since  fx  and  /2  are  prime  to  each  other.  We 
have  then 

fk-2  =  qk-2fk-1  +fk. 

Putting  now  the  value  of  /3  from  i)  in  2)  we  have 

/4=(i+2i?2)/2  —  qJi', 
that  is  L  =  r1f1  +  r2f2, 

where  r,  and  r2  are  polynomials  in  x.  Putting  the  expressions  for 
/3  and  f4  in  terms  of  fx  and  f2  in  3),  we  obtain 

T 5  ===  ^1/1  T  ^2/  2> 

where  f^  s2  are  polynomials  in  ^r.  Continuing  this  process,  we 
obtain  finally 

fk  =  W1f1  +  ZV2f2, 

where  wlt  w2  are  polynomials  in  x.  As  has  been  said,  /&  is  a  con- 
stant different  from  o.     Putting  therefore 

wt  =  A-4>i  O) ,     Wa  =  fk<l>2  (x) , 


270       GENERAL   THEOREMS    CONCERNING   ALGEBRAIC    NUMBERS. 

we  have 

ftfc-(*)A"(*)  +/i*i(*)/iOO  =/*» 

and  hence 

where  ^(x)  and  <£2(.*")  are  polynomials  in  x. 

We  may  generalize  the  above  theorem  as  follows: 

Theorem  2.  //  A  (■*')  an^  f*  (x )  ^  *wo  polynomials  in  x  without 
a  common  divisor  and  g{x)  any  polynomial  in  x,  there  exist  two 
polynomials  in  x,  ^(a*)  and  $2(x),  such  that  ®2(x)  w  of  lower 
degree  than  fx(x)  and 

•kWAW  +  •*(*)/.(*)  — tft*)'i 

By  Th.  2  there  exist  two  polynomials  in  x,  ^(x),  </>2(a),  such 

that  ^)/iW+fc(')/iW  =  i  4) 

Multiplying  4)  by  ^(^)  we  have 

#0)4>iO)/iO)  +#<*>**(•*)/*(*)  =#<»•        5) 

Putting  #(a-)<£2(a")  m  the  form 

^W^W=?W/iW  +  '(*)» 

where  #(a)  and  r(.r)  are  polynomials  in  x  and  r(.r)  is  of  lower 
degree  than  f1(x),  and  substituting  in  5),  we  have 

[g^)<f>1(x)+q(x)f2(x)]f1(x)+r(x)f2(x)=g(x); 

that  is  *d*)fd*)+*t*)fn(*)=9(*)* 

where  $x{x)  and  ®2{x)  are  polynomials  in  x,  and  ®2(x)  *s  °f 
lower  degree  than  fx{x). 

A  polynomial,  /(a),  is  said  to  be  irreducible  in  the  realm 
k(a)  when  it  cannot  be  resolved  into  integral  factors  whose  coeffi- 
cients are  numbers  of  k(a).  When  f(x)  has  rational  coefficients 
and  is  said  simply  to  be  irreducible,  no  realm  being  specified,  the 
rational  realm  is  understood;  that  is,  f(x)  is  not  resolvable  into 
integral  factors  having  rational  coefficients. 

Theorem  3.  An  irreducible  polynomial,  f(x),  can  have  no 
factor  in  common  with  another  polynomial,  F(x),  unless  F(x) 
be  divisible  by  f(x). 


GENERAL   THEOREMS    CONCERNING   ALGEBRAIC    NUMBERS.        27 1 

The  coefficients  of  the  greatest  common  divisor  of  the  two 
polynomials  F(x)  and  f(x)  are  derived  from  the  coefficients  of 
these  two  polynomials  by  rational  operations  and  are  therefore 
rational  numbers,  since  the  coefficients  of  F(x)  and  f(x)  are 
rational  numbers. 

But  f(x)  is  divisible  by  no  polynomial  in  x  with  rational  coeffi- 
cients except  itself  and  the  rational  numbers.  Hence  either  F(x) 
and  f(x)  have  no  common  factor  or  F(x)  is  divisible  by  f(x). 

Cor.  1.  If  f(x)  be  irreducible  and  F(x)  vanish  for  one  root 
of  the  equation  f{x)  —o,  it  vanishes  for  all  roots  of  f(x)  =o. 
For,  if  F(x)  vanish  for  a  root  of  f(x)  =o,  F(x)  and  f(x)  must 
have  a  common  factor.     But  this  can  only  be  f(x). 

Cor.  2.  If  f(x)  be  irreducible  and  F(x)  be  a  function  of 
lower  degree  than  f(x)  that  vanishes  for  one  root  of  f(x)  =o, 
then  F(x)  must  vanish  identically;  that  is,  all  coefficients  of 
F(x)  are  o. 

§  2.    Numbers  of  a  Realm. 

Let  us  consider  the  realm  k(a)  of  the  nth  degree,  a  being  a 
root  of  the  irreducible  rational  equation 

/O)  =  x-  +  a,***  H h  *»=0,  1 ) 

whose  remaining  roots  we  denote  by  a', a",  •••,a(n"1). 

Any  number  0  of  k(a),  being  produced  from  a  by  repeated 
performance  of  the  operations  of  addition,  subtraction,  multipli- 
cation and  division,  is  a  rational  function  of  a  with  rational  coeffi- 
cients and  hence  can  be  expressed  in  the  form 

where  x(a)  and  «A(a)  are  rational  integral  functions  of  a  with 
rational  coefficients.  The  realm  k(a)  is  composed  therefore  of 
all  rational  functions  of  a  with  rational  coefficients,  the  denomi- 
nator never  being  o. 

We  shall  now  show  that  every  number  of  the  realm  can  be 
expressed  as  a  rational  integral  function  of  a  with  rational 
coefficients. 


272       GENERAL  THEOREMS   CONCERNING  ALGEBRAIC   NUMBERS. 

The  degrees  of  x(a)  and  $(&)  can  be  made  lower  than  the 
nth  by  virtue  of  the  relation 

an  +  a^-1  -\ f-  an  =  o. 

Since  ^(a)  1S  different  from  o  and  of  degree  lower  than  the  nth, 
\p(x)  is  not  divisible  by  f(x),  and  hence,  since  f(x)  is  irreducible, 
if/(x)  is  prime  to  f(x)  (Th.  3).  We  can  therefore  by  Th.  1  find 
two  polynomials  in  x,  ^1(x),  &2(x),  with  rational  coefficients  and 
<£2(.r)  of  lower  degree  than  the  wth,  such  that 

$1W/M+$2(^W=XW.  2) 

Putting  a  for  x  in  2)  we  have 

**(a)y(<x)—  x(a), 

and  hence 

that  is,  0  =  &0  +  &xa  +  fr2a2  -] j-  fc^a*"1, 

where  bQ,bx,  •••ibn_x  are  rational  numbers.  This  representation 
of  0  is  unique,  for,  if  we  had  also 

0  =  c0  +  c%a  +  c2a2  H l.  Cn^a"-1, 

then  it  would  follow  that 

fro  — ^0+  (fri  —  0<H h  (frn_x  —  cn_1)an-1  =  o; 

that  is,  a  polynomial  in  ^r  of  degree  lower  than  the  nth  would 
vanish  for  x  =  a,  but  this  by  Th.  3,  Cor.  2  is  impossible  unless  all 
the  coefficients  of  the  polynomial  are  o.     Hence 

and  the  two  representations  are  identical. 

The  numbers  of  the  realm  are  seen  therefore  to  be  coextensive 
with  the  totality  of  rational  integral  functions  of  a  with  rational 
coefficients  and  of  degree  not  higher  than  the  (w  —  i)th. 

We  shall  next  prove  the  following  simple  theorem : 


GENERAL   THEOREMS   CONCERNING   ALGEBRAIC   NUMBERS.        2/3 

Theorem  4.  Every  number  6  of  k(a)  satisfies  a  rational 
equation,  whose  degree  is  the  same  as  that  of  the  realm,  and 
whose  remaining  roots  are  the  conjugates  of  0. 

Form  the  equation 

<!>(t)  =  (t  —  0)(t  —  0')'--(t  —  6<n-1>) 

=  tn  +  ditn-l+...+  dn==0)  3) 

where  0',  0",  •  •  •,  #***  are  the  conjugates  of  6. 

The  coefficients,  dlf  d2,  •••,  dn,  of  3)  are  symmetric  functions  of 
the  roots  of  1)  and  hence  rational  functions  of  the  coefficients 
of  1).  Hence  d±,d2,  -~>dn  are  rational  numbers.  Therefore  $ 
satisfies  a  rational  equation  of  the  nth  degree,  whose  remaining 
roots  are  the  conjugates  of  6.  Every  number  of  the  realm  is 
therefore  evidently  an  algebraic  number. 

We  turn  now  to  the  reducibility  of  $(0>  and  shall  prove  the 
following  theorem : 

Theorem  5.  The  function  <£(£)  is  either  irreducible  or  is  a 
power  of  an  irreducible  function.  The  n  conjugates  of  a  number 
of  k(a)  are  either  all  different  or  else  fall  into  nx  systems,  each 
containing  n2  numbers  all  alike.  In  the  first  case,  &(t)  is  irre- 
ducible, in  the  second,  &(t)  is  th£  nxth  power  of  an  irreducible 
function  of  the  n2th  degree. 

If  &(t)  be  reducible  it  must  be  a  product  of  irreducible  factors, 
each  of  which  vanishes  for  one  or  more  of  the  quantities 

0,6',  ••.,0<n-1>. 

Let  *(0=*i(0*2(0---*»i(0> 

where  <j>x{t) , $2{t) ,  •••,<^>m(0  are  irreducible  and  let  <f>x{t)  vanish 
for  t  =  0 ;  that  is, 

<f>1(0)=o. 

We  have  seen  that 

o=g(a), 

where  a  is  the  number  defining  the  realm  and  g(a)  a  rational 
integral  function  of  a  with  rational  coefficients.     Then 

<£i[#(s)]=o. 
18 


274       GENERAL   THEOREMS    CONCERNING   ALGEBRAIC    NUMBERS. 

The  equations 

4>i[#0')]=o  and  /(*)=0 

have  therefore  a  root  in  common,  and,  since  f(x)  is  irreducible, 
<f>i[9(x)]  must  vanish  for  all  roots  of  f{x)  =o;  that  is, 

^[^(^)]=o,^1[^(a-)]=o,...,^1[5r(a^)]. 

But       0'  =  g(a'),e"  =  g(a"),.->,0^=g(a(n-v). 

Hence 

$,(6)  =0,^(6')  =o,  -..^^"-v)  =o; 

that    is,   £i(0    vanishes    for   all   of   the    w   conjugate   numbers 

0,0',  •••,0(n_1). 

If  these  numbers  be  all  different,  <f>t(t)  is  of  the  nth  degree 
and  hence  identical  with  $(0- 

If,  however,  there  be  among  them  only  n2  which  are  different 
from  each  other,  say 

6,0',  ...,0<"«-1>, 

then  <j>1(t)  =  (t  —  6)(t  —  6f)  •••  (t  —  0^-^). 

Since,  moreover,  every  irreducible  factor  of  ®(t)  vanishes  for 
one  of  the  quantities  0, 6',  •••,  0W_1,  and  hence  for  all  of  them 
(Th.  3,  Cor.  2),  every  one  of  these  irreducible  factors  of  &(t)  is 
identical  with  <f>r(t)  ;  that  is  ^(Oj^sCO*  •**»^n(0  are  all  iden- 
tical with  ^(0- 

Therefore  $(t)  is  in  this  case  a  power  of  <f>x(t)  ;  that  is, 

$(t)  =  [<^>i(0]nS  where  %n2  =  n. 

We  have  seen  (Chap.  I,  §  i)  that  every  algebraic  number  sat- 
isfies a  single  irreducible  rational  equation. 

We  see  now  from  the  above  that  the  degree  of  this  equation 
is  a  divisor  of  the  degree  of  the  realm  of  which  6  is  a  number. 
According  as  the  degree  of  this  equation  is  the  same  as  or  lower 
than  that  of  the  realm,  6  is  said  to  be  a  primitive  or  imprimitive 
number  of  the  realm. 

Thus  6  is  a  primitive  number  of  k(a)  when  ft  is  different  from 
all  of  its  conjugates  and  an  imprimitive  number  when  this  is  not 
the  case. 


GENERAL  THEOREMS   CONCERNING   ALGEBRAIC   NUMBERS.        275 

Theorem  6.  Any  primitive  number  0  of  k(a)  may  be  taken 
to  define  the  realm;  that  is, 

k(0)  =  k(a). 

Let  0  be  any  primitive  number  of  k(a)  and  $',  0",  •••,0(n_1)  its 
conjugates,  and  let  w  be  any  number  of  k(a)  and  <«/, <d",  •••,o)(n~1) 
its  conjugates.  We  shall  show  that  o>  can  be  expressed  as  a 
rational  function  of  0  with  rational  coefficients,  and  hence  that 
k(0)=k(a). 

We  have 

q>(t)  =  (t  —  0)(t—0')  •••  (*—  ^(n"1)). 
Then 

(ft)  ft)'  G)(n_1)       \ 

7zrg  +  — 3-, +  ...  +  7-^r))  _*(,).  4) 

where  ^(O  is  a  polynomial  in  £  of  the  (« —  i)th  degree,  whose 
coefficients  are  rational  numbers,  for  they  are  symmetric  func- 
tions of  the  roots  of  the  irreducible  rational  equation  satisfied  by 
a,  and  hence  rational  functions  of  its  coefficients.  Putting  0  for 
t  in  4)  we  have 

<»(0  —  0f){0  —  0")  ■•.(0  —  0<n-v)=*(0), 
or,  putting  as  usual 

d/dt>$>(t)=&(t)  =  (t  —  0'){t  —  0").-.(t  —  d<"-1>)+terms. 
containing  the  factor  t  —  0,  we  have 

where  &(0)  is  a  polynomial  in  t  with  rational  coefficients,  and  is 
different  from  o,  since  0  is  different  from  all  its  conjugates. 
Every  number  of  k(a)  can  therefore  be  expressed  as  a  rational 
function  of  0  with  rational  coefficients.  Hence  all  numbers  of 
k(a)  are  numbers  of  k(0),  and  therefore 

k(a)=k(0). 

Theorem  7.    If  f(x) =**  +  axxn~x  ^ f-  an  =  0  5 ) 

be  an  irreducible  rational  equation,  and  0,  one  of  its  roots,  be  an 


276       GENERAL  THEOREMS   CONCERNING  ALGEBRAIC   NUMBERS. 

algebraic  integer,  the  remaining  roots,  0',  0",  •••,  0(n_1),  are  also 
algebraic  integers. 

This  theorem  follows  directly  from  Th.  4,  Chap.  II.  It  may 
also  be  proved  as  follows. 

Since  0  is  an  integer,  it  must  satisfy  an  equation 

F(x)  =  xn  +  bxx**  H h  ^  =  o,  6) 

whose  coefficients  are  rational  integers.  But  if  F(x)  vanish  for 
one  root  of  the  irreducible  equation  5),  it  vanishes  for  all  roots 
of  5).     Hence  6',  6",  •  ••,0(n-1)  satisfy  6)  and  are  integers. 

Theorem  8.  The  sum,  difference,  product  and  quotient,  the 
denominator  of  the  latter  not  being  zero,  of  two  algebraic  num- 
bers are  algebraic  numbers. 

Let  a  and  /?  be  two  algebraic  numbers,  which  satisfy  respect- 
ively the  two  irreducible  rational  equations 

xm  +  a^™-1  -\ h  am  =  0,  7) 

,*•  + fr^H \-bn  =  o.  8) 

The  necessary  and  sufficient  condition  that  a  +  (3  shall  be  an 
algebraic  number  is  that  it  shall  satisfy  a  rational  equation. 
Form  the  equation 

[x—(a  +  p)]  •••  [(*— .(a«>+0<»)]  •••  [*—  (a(«-4)+j8<«-1>>] 

=Xmn  +  ClXmn^  H \-Cmn  =  0,  9) 

whose  roots  are  the  mn  numbers 


f  a  =  a,a',  •••,a(w-1), 
a  +  £'  J  /3  =  p,(3',---,/3<n-1\      ■ 


The  coefficients  clt  c2,  •••,cmn  of  9)  are  symmetric  functions  of 
the  roots  of  7)  and  8),  and  hence  rational  functions  of  the  coeffi- 
cients of  7)  and  8). 

But  the  coefficients  of  7)  and  8)  are  rational  numbers. 

Hence  the  coefficients  of  9)  are  rational  numbers,  and  a-\-/3 
is  therefore  an  algebraic  number.  The  proofs  for  a  —  /?,  a/3  and 
a/f3  are  of  the  same  character. 

Cor.  1.  Every  rational  function  of  any  number  of  algebraic 
numbers  with  rational  coefficients  is  an  algebraic  number. 


GENERAL   THEOREMS    CONCERNING   ALGEBRAIC   NUMBERS.        277 

Cor.  2.  The  sum,  difference  and  product  of  two  algebraic  in- 
tegers are  algebraic  integers;  for  in  this  case  the  c's  being  not 
only  rational  but  integral  functions  of  the  a's  and  b's,  and  the  a's 
and  b's  being  now  integers,  the  c's  are  themselves  rational  integers. 

Cor.  3.  Every  rational  integral  function  of  any  number  of 
algebraic  integers  with  rational  integral  coefficients  is  an  algebraic 
integer. 

We  obtain  a  still  more  general  theorem  when  we  notice  that,  if 
we  allow  the  coefficients  bx,  b2,  --,bn  of  the  equation 

X*  +  ^«-i  _| (_  0n  =  0  IO) 

to  be  any  algebraic  numbers  instead  of  restricting  them  to  rational 
numbers,  the  roots  of  10)  will  nevertheless  be  algebraic  numbers. 

Theorem  9.     If  &  be  a  root  of  the  equation 

F(x)=.rn  +  a1xn-i-{ \-an  =  o, 

where  ax,a2,--,an  are  any  algebraic  numbers,  it  is  itself  an 
algebraic  number. 

Let  ax, a2,  '-,an  satisfy  rational  equations  of  degree mx, m2,  •  •  •, 
mn,  respectively,  and  let  the  remaining  roots  of  these  equations  be 

a  '  a  "   ...   rt  t*"*-1) 

Let  m  —  mxm2  •••  mn  and  form  by  putting  for  a<  a*,  a/,  •••, 
a(mi-i)  (i—  i}  2,  ••-,  n)  the  m  polynomials  in  x 

F(x)      =xn  +  axxn-*  H h  an, 

Fx(x)     =xn  +  ax'xn-i  H \-an, 

F2(x)     =xn  +  a/ V-1  H h  a», 


Fm_x(x)  =xn  +  a/^-1^"-1  H h  a„(m»-1). 

Form  the  product 

FFxF2.-.Fm_x  =  f(x). 


278        GENERAL   THEOREMS    CONCERNING   ALGEBRAIC   NUMBERS. 

The  coefficients  of  f(x)  will  be  symmetric  functions  of  the 
roots  of  the  rational  equations  satisfied  by  a19  a2,  •  •  • ,  an,  and 
hence  rational  functions  of  their  coefficients.  They  are  therefore 
rational  numbers  and  <o,  being  a  root  of  the  rational  equation 

is  an  algebraic  number. 

Ex.  1.    Let  w  be  a  root  of  the  equation 

F(x)  =  x2  +  V  2*  +  y/J=  o.  11) 

We  see  that  V2  and  V3  are  roots  respectively  of  the  rational  equations 

x2  —  2  =  0      and      x'1  —  3  =  o, 

whose  remaining  roots  are  —  V2  and  —  V3-    We  have 

FxO)  =  x2  +  \/2X  —  V3T 

F2  O)  =  x2  —  \fex  +  V3> 

F3(x)  —x2  —  yj2x  —  V3~i 

and  f(x)  =F  F!F2Fz  =  xs  —  4x*  —  2xi — 12^  +  9  =  0  12) 

Hence,  w  being  a  root  of  12),  is  an  algebraic  number.  It  is  moreover  an 
integer,  since  the  coefficients  of  11)  are  integers  (see  Cor.  1  below). 

Cor.  I.     If  to  be  a  root  of  the  equation 

F(x)=xn  +  a1xn~1+  •••  -j-an  =  o, 

where  alta2,  '-,an  are  algebraic  integers,  it  is  itself  an  algebraic 
integer;  for  the  coefficients  of  f(x)  formed  as  above  are  not  only 
rational  but  integral  functions  of  the  coefficients  of  the  rational 
equations  satisfied  by  the  a's  and  these  are  now  rational  integers. 
Hence  the  coefficients  of  f(x)  are  rational  integers,  and  o>  is  an 
integer. 

Theorem  10.  Every  algebraic  number  can  by  multiplication 
by  a  suitable  rational  integer  be  made  an  algebraic  integer. 

Let  the  algebraic  number,  a,  be  a  root  of  the  rational  equation 

and  let  a0  be  the  least  common  denominator  of  the  a's.     Then 
an  +  -J  •  a71'1  +  -2  •  a11-2  +  .  .  .  +  -=  =  o,  13) 

where  the  b's  are  rational  integers. 


GENERAL   THEOREMS   CONCERNING  ALGEBRAIC   NUMBERS.        279 

Multiplying  13)  by  a0n,  we  have 

(a.aY  +  ^(0,0.)^  +  aQb2(a0a)n-*  +'••  +V-1&»  =  o; 
that  is,  a0a  is  a  root  of  the  equation 

yn  +  bj*-1  +  a0b2yn-*  H f-  ao"-^,  =  o, 

whose  coefficients  are  rational  integers,  and  is  therefore  an  alge- 
braic integer. 
Ex.    Let  a  be  a  root  of 

•*3-hf*2  +  f*  +  !  =  o, 

that  is,  of  *  +  TV2.+  H*  +  tI  =  o.  J4) 

Multiplying  14)  by  123,  we  have 

(i2x)3  -\-  6(i2x)2  -\-  192(12*)  +2160  =  0. 
Thus  12a  is  a  root  of  the  equation 

y3  +  6y2  +  192V  +  2160  =  o, 
and  hence  an  integer. 

This  is  seen  to  be  simply  the  transformation  of  13)  into  an 
equation  whose  roots  are  a  times  those  of  1),  a  being  selected 
so  as  to  make  the  coefficients  of  the  new  equation  integers. 


CHAPTER   X. 
The  General  Quadratic  Realm. 

§  i.    Number  Defining  the  Realm. 

By  the  general  quadratic  realm  we  understand  the  realm  de- 
fined by  a  root  of  the  general  irreducible  quadratic  equation  of 
the  form 

ax2  +  bx  +  c  =  o,  I ) 

where  a,  b  and  c  are  rational  integers. 

If  a  be  a  root  of  i),  this  realm  is  denoted  by  k(a).  If  a'  be 
the  other  root  of  i),  the  realm  k(a')  is  the  conjugate  realm  of 
fc(a)(Chap.  I,  §4). 

Solving  i),  we  have 


—  b  +  V&  —  4ac  ,       —  b  —  V  b*  —  ^ac 

a  = ,      a   = 

2a  2a 

Put  b2  —  4ac  =  l2m, 

where  m  contains  no  square  factor ;  then 


V&2- 

—  4ac  =  iym, 

and 

k{a)  =k(ym) 

for 

a 

—  b  -}-  /  V  m 
2a 

is  evidently  a 

number  of 

fc(Vm)  and 
—      2aa  -f  b 

\/  *n : 

/ 

is  a  number  of  k(a). 
Hence  k{a)=k{ym).1 

Hence,  to  consider  all  quadratic  realms,  it  is  sufficient  to  con- 
sider all  realms  defined  by  a  root  of  an  equation  of  the  form 

x2  —  m  =  o,  2) 

1  See  Chap.  IX,  Th.  6. 

280 


THE   GENERAL    QUADRATIC   REALM.  28 1 

where  m  is  any  rational  integer  containing  no  squared  factor. 
We  shall  understand  in  what  follows  by  ~\Jm  the  positive  real  or 
imaginary  root  of  2),  and  shall  assume  that  m  contains  no  square 
factor. 

The  conjugate  realms  k(a)  and  k(a')  are  identical,  since  a  is 
evidently  a  number  of  k(a')  and  a'  a  number  of  k(a). 

The  general  quadratic  realm  is  the  simplest  example  of  what  is 
known  as  a  Galois  realm;  that  is,  one  which  is  identical  with  all 
its  conjugate  realms. 

§2.  Numbers  of  the  Realm.  Conjugate  and  Norm  of  a 
Number.    Primitive  and  Imprimitive  Numbers. 

Let  a  be  a  root  of  the  irreducible  quadratic  equation 

*2  +  Px  +  Q.  —  °- 
Every  number,  w,  of  k(a)   is  a  rational  function  of  a  with 
rational  coefficients,  and  hence  has  the  form 

a  -f  ba 

(0=z  — , 

c  +  a a 
where  a,  b,  c  and  d  are  rational  numbers. 

a  -f  ba 


The  number  co'  = 


c  +  do! 


obtained  from  <o  by  the  substitution  of  a!  for  a  is  the  conjugate 
of  w  (Chap.  I,  §  4).  The  numbers  of  k(a)  that  are  rational  are 
seen  to  be  their  own  conjugates.  We  shall  show  now  that  every 
number,  w,  of  k{a)  can  be  put  in  the  form 

co  =  e  +  fa* 

where  e  and  /  are  rational  numbers.1 
First,  let  abe  -\/m.     Then  we  have 


c  +  dV 


1) 


m 


"See  Chapter  VIII,  §2,  for  general  theorem  of  which  this  is  a  special 
case.  Simplified  proofs  are  given  here  of  this  and  several  following 
theorems. 


282  THE   GENERAL   QUADRATIC   REALM. 

Multiplying  the  numerator  and  denominator  of  i)  by  c — d^Jm, 
we  obtain 

ac  —  bdrn       be  —  ad     ,— 

~~  c2  —  d2m       c2  —  d2m 

All  numbers  of  k(\/m)  can  therefore  be  put  in  the  form 
e  +  /V  ra,  where  e  and  /  are  rational  numbers. 

If  w,  =  a-[-frVw>  be  any  number  of  k(-\/m)  it  satisfies  the 
quadratic  equation 

x2 —  2ax-\-a2  —  mb2  =  o,  2) 

whose  other  root  is  uy',  =  a — b^Jm,  the  conjugate  of  w.  Hence 
every  number  w  of  k(^/m)  satisfies  a  rational  equation  of  the 
second  degree  (Chap.  IX,  Th.  4).  We  say  that  a  is  a  primitive 
or  imprimitive  number  of  k(^/m)  according  as  the  equation  2) 
is  irreducible  or  reducible. 

The  necessary  and  sufficient  condition  for  2)  to  be  irreducible 
is  evidently  b=^=o.  In  other  words,  a  is  a  primitive  number  if  it 
be  different  from  its  conjugate  (Chap.  IX,  Th.  5). 

If  b  =  o,  and  hence  w  =  w'  =  a,  then  w  satisfies  the  rational 
equation  of  the  first  degree 

x  —  0  =  0. 

The  primitive  numbers  of  a  realm  are  thus  seen  to  be  those 
defined  by  equations  of  the  same  degree  as  that  of  the  realm 
(Chap.  IX,  Th.  5).  The  imprimitive  numbers  of  a  quadratic 
realm  are  evidently  the  rational  numbers. 

If  «  be  a  primitive  number  of  a  realm  of  the  wth  degree  and  the 
identity 

Go  +  ai«  -j (-  an-iw"-1  =  b0  +  bi<a-\ +  frn-iW*"1  3) 

exist  where  the  a's  and  b's  are  rational  numbers,  then  the  coefficients  of 
the  same  powers  of  «  in  the  two  members  of  3)  must  be  equal;  that  is, 

a0=bo,    di  =  bi,     •••,    a»-i  =  &n-i ; 

for  otherwise  «  would  satisfy  an  equation  of  degree  lower  than  the  wth, 
which  is  contrary  to  the  assumption  that  w  is  a  primitive  number  of  the 
realm. 

We  have  shown  (Chap.  IX,  Th.  6)  that  any  algebraic  number 
realm  can  be  defined  by  any  one  of  its  primitive  numbers.     This 


THE    GENERAL   QUADRATIC   REALM.  283 

can  be  proved  for  the  special  case  of  quadratic  realms  very  simply 
as  follows: 

Let  a  be  a  primitive  number  and  w  any  number  of  &(\/m). 
We  have  seen  above  that  a  and  w  can  be  put  in  the  forms 

a  =  a-\-b-\/m,  4) 

(o  =  c  +  dy/m,  5) 

where  a,  b,  c  and  d  are  rational  numbers. 


From  4)  we  have 

,—      a  —  a 
1/111  — 

Vm~      b     ' 

and  from  5) 

be  —  ad      d 
—        b        +-da. 

Hence  every  number  <o  of  k(^/m)  can  be  written  in  the  form 
w  =  e  +  fa, 
where  e  and  /  are  rational  numbers  and  a  a.  primitive  number  of 
fc(Vw)-     Hence 

and  we  have  proved  not  only  that  every  quadratic  realm  may  be 
defined  by  any  one  of  its  primitive  numbers,  a,  but  that  every 
number,  w,  of  the  realm  k(a)  may  be  put  in  the  form 

<1>  =  e  +  fa, 

where  e  and  /  are  rational  numbers  (Chap.  IX,  §  2). 

We  may  evidently  choose  as  the  primitive  number  defining  the 
realm  an  integer.  In  what  follows  we  shall  suppose  this  to  have 
been  done.  The  product  of  a  number,  w,  of  k(a)  by  its  con- 
jugate a/  is  its  norm1  and  is  denoted  by  «[<u]  ;  that  is, 

w[to]  =toto'. 

Since  n[w]  is  a  symmetric  function  of  the  roots  of  the  rational 
equation  satisfied  by  a,  it  is  a  rational  function  of  the  coefficients 
of  this  equation,  and  hence  a  rational  number.  In  particular 
when  the  realm  is  defined  by  V  w,  we  have 

w[to]  =  (a-\-b-\/m)(a  —  b}/m)  =a2  —  b2m. 

'Hilbert:  Bericht,  §3. 


284  THE   GENERAL    QUADRATIC   REALM. 

§3.    Discriminant  of  a  Number.1 

The  square  of  the  difference  of  a  number  a  and  its  conjugate 
is  called  the  discriminant  of  the  number  and  is  denoted  by  d[a]  ; 
that  is, 

^[a]  ==(<*- a')2  = 

It  is  evidently  a  rational  number  and  the  discriminant  of  the 
quadratic  equation 

x2  +  axx  +  a2  =  o, 

whose  roots  are  a  and  a'. 

If  a  be  a  primitive  number  of  the  realm  its  discriminant  is 
different  from  o,  and  conversely,  if  d[a]  be  different  from  o,  a 
is  a  primitive  number. 

§  4.    Basis  of  a  Quadratic  Realm. 

Theorem  i.  There  exist  in  every  quadratic  realm  two  in- 
tegers, »!,  w2,  such  that  every  integer,  <o,  of  the  realm  can  be 
expressed  in  the  form 

(a  =  a1o)1  -j-  a2a)2, 

where  axa2  are  rational  integers.2 

Suppose  the  realm  to  be  defined  by  an  integer,  a,  a  supposition 
in  no  way  limiting  the  generality  of  the  proof,  and  let  w  be  any 
integer  of  k(a).  By  the  preceding  paragraph  w  can  be  put  in 
the  form 

<»  =  r1  +  r2a,  1) 

where  rt  and  r2  are  rational  numbers.     We  have 

o>  —rx-\-r2a!.  2) 

Solving  1)  and  2)  for  rx  and  r2  by  means  of  determinants, 
we  have 

'Hilbert:  Bericht,  §3. 

2  Hilbert :  Bericht,  Satz  5.  This  proof  could  have  been  somewhat  sim- 
plified had  greater  use  been  made  of  the  fact  that  the  realm  under  con- 
sideration was  quadratic,  but  it  seemed  desirable  to  give  the  proof  in  a 
form  at  once  extendable  to  realms  of  any  degree. 


THE   GENERAL   QUADRATIC   REALM. 


285 


r,  = 


ft) 

a 

1  w 

a 

|I      a\ 

ft)' 

a! 

|v 

a' 

|I     «'l 

I 

a 

I1 

-I2 

I 

a! 

1  1 

a' 

I 

ft) 

1    I 

a 

I 

ft)' 

l« 

a' 

1    I 

a 

2 

|1 

a' 

*M' 


d\ay 


where  Ax  and  ^42  are  rational  integral  functions  of  the  integers 
<o,  a,  a  and  a'  with  integral  coefficients  and  hence  integers  (Chap. 
IX,  Th.  8,  Cor.  3). 

Moreover,  d [a]  is  a  rational  number  and  hence  Alf  =  r1d[a], 
and  A2i  =  r2d[a],  are  rational  numbers.  Therefore,  Ax  and  A2 
are  rational  integers.  Hence  every  integer,  o>,  of  k[a]  can  be 
put  in  the  form 

A  +  A2a 

where  ^t  and  ^42  are  rational  integers  and  J[a]  is  the  discrimi- 
nant of  a. 

Suppose,  now,  all  integers  of  the  realm  to  be  written  in  the 
form  3)  and  consider  those  in  which  A2  is  not  equal  to  o. 
Among  these  there  will  be  some  in  which  A2  will  be  smaller  in 
absolute  value  than  in  any  of  the  remaining  ones. 

Ax'  +  A2'a 


3) 


Let 


ft>„  = 


rf[a] 


be  one  of  these.  Then  A2  will  be  the  greatest  common  divisor 
of  the  values  of  A2  in  all  integers  of  the  realm;  for  if  this  be  not 
the  case,  let 

Ax"  +  A2"a 

be  any  integer  such  that  A"  is  not  divisible  by  A2,  and  let  A  be 
the  greatest  common  divisor  of  A2  and  A2" .  Then  we  can  find 
two  rational  integers  a  and  b  such  that 

aA2'  +  bA2"  =  A,   I       . 


286  THE    GENERAL   QUADRATIC    REALM. 

and  hence 

7  =  ™2  +  ^3  = -^ 

aA/  +  6At"  +  Aa 

is  an  integer  in  which  the  coefficient  of  a  is  less  in  absolute  value 
than  A2,  which  is  contrary  to  the  supposition  that  there  is  no 
value  of  A 2  less  in  absolute  value  than  A2'.     Hence 

A2  =  a2A2 , 

where  a2  is  a  rational  integer. 

Denoting  «  —  a2w2  by  to*,  we  have 

Ax  +  A2a  -  <y4/  -  a2A2fa      Ax  -  a2A/ 
d\a\  d[a]        ' 

Consider  now  those  integers  of  the  realm  in  which  A2  =  o, 
butA^o.1 

There  will  be  one  or  more  among  them  in  which  Ax  is  less  in 
absolute  value  than  in  any  of  the  remaining  ones. 

Let  <o1  =  A1,,,/d[a] 

be  one  of  them.  We  see  as  above  that  A™  is  the  greatest  com- 
mon divisor  of  the  values  of  At  in  all  the  integers  in  which 


A2  =  o,    ^/4=o,2 

and  hence 

<o*  =  to  —  a2o)2  =  d^x, 

or 

0)  =  Ojtoi  -j-  t72co2. 

4) 

There  exist,  therefore,  in  every  quadratic  realm  two  integers, 
<alf  eo2,  such  that  every  integer,  w,  of  the  realm  can  be  expressed  in 
the  form  4),  when  alt  a2  are  rational  integers. 

^he  remainder  could  be  worded  much  more  simply,  if  the  fact  that 
(At —  0^4/) /d [a]  is  a  rational  integer  be  made  use  of,  but  the  above  form 
seems  better  as  it  is  in  line  with  the  general  theorem. 

2 The  integers,  in  which  A*=zo  and  Ai=^=o,  are  evidently  the  rational 
integers,  0  excluded.  Also  A"'  =  d[a],  and  wx— 1.  We  have 
Ax —  a2Ai  =  a^i",  where  at  is  a  rational  integer. 


THE   GENERAL   QUADRATIC   REALM.  287 

Every  pair  of  integers,  wls  »3J  possessing  this  property  is  called 
a  basis  of  k(a). 

Cor  1.  //  »XI  w2  be  a  basis  of  k(a),  then  w/,  «2'  is  a  basis  of 
the  conjugate  realm  k(a'). 

Theorem  2.     If  mlf  w2  be  a  basis  of  k(-\/m),  the  necessary  and 
sufficient  condition  that 
•  *  1 

W2*  =  b^-L  +  ^2W2> 

w/^r*?  alf  a2,  &1?  &2  are  rational  integers,  shall  be  also  a  basis  of 
k(^/m)  is 


K    K 


—  db  I, 


For  the  proof  of  this  theorem  see  the  corresponding  one  in 
k(i)  (Chap.  V,  Th.  1). 

§  5.    Discriminant  of  the  Realm. 

If  m19  w2  be  a  basis  of  &(\/m),  the  square  of  the  determinant 
formed  by  these  integers  and  their  conjugates  is  called  the  dis- 
criminant of  the  realm  and  is  denoted  by  d;  that  is, 

13 


*-P    "' 


2 


We  see  that  d  is  a  rational  integer,  for  it  is  an  integral  sym- 
metric function  of  the  roots,  V w,  — y/m,  of  the  equation 

x2  —  m  =  o, 

and  hence 'a  rational  integral  function  of  the  coefficients  of  this 
equation,  which  are  rational  integers. 

That  the  value  of  d  is  independent  of  the  basis  chosen  may  be 
shown  as  in  k(i). 

The  discriminant  of  every  integer  of  the  realm  is  divisible  by 
the  discriminant  of  the  realm;  for,  if 


a  =  a1(o1  -j-  a2o)2, 


"Hilbert:  Bericht,  p.  181. 
2Hilbert:  Bericht,  p.  194. 


288 


THE   GENERAL   QUADRATIC    REALM. 


be  any  integer  of  k(^/m),  and 

i  =  b1o>1  -J-  b2a>2, 


then 


d\a\  = 


K 

w\ 

al  at\ 

=  ^V. 

d[a]=d, 

K   K 

=  d= 

al 

a2 

CO, 


If 

then 


and  i,  a,  is  a  basis  of  the  realm. 

We  see,  moreover,  that  when  d [a]  is  not  divisible  by  the  square 
of  a  rational  integer,  we  have    Q 

d[a]=d, 
and  hence  I,  a,  is  a  basis.1 

The  converse  of  this  theorem  is,  however,  not  true;  that  is 
d[a]  may  be  divisible  by  the  square  of  a  rational  integer  and 
still  i,  a,  be  a  basis. 

1  The  definition  and  deductions  of  this  paragraph  are  immediately  ex- 
tendable to  the  general  algebraic  realm  of  the  wth  degree.  The  last  fact 
mentioned  is  of  especial  importance  as  it  may  be  shown  by  the  method 
used  in  the  text  that,  if  0  be  a  root  of 

Xn  +  diXn~x  -\ +  an  =  o, 

where  au  •••,  an  are  rational  integers,  and  d[0]  be  not  divisible  by  the 
square  of  a  rational  integer,  then  I,  #,  ...,  0n_1  is  a  basis  of  k(B).  The 
great  value  of  this  fact  is  that  although  we  may  by  the  method  of  §  4  prove 
the  existence  of  a  basis  in  a  realm  of  the  nth  degree,  we  have,  however, 
general  methods  of  determining  a  basis  only  in  the  cases  of  »  =  2  or  3. 
The  case  n  =  2  will  be  discussed  in  the  next  paragraph ;  that  for  n  =  3  will 
be  found  in  Woronoj :  The  Algebraic  Integers  which  are  Functions  of  a 
Root  of  an  Equation  of  the  Third  Degree,  this  being  a  translation  of  the 
Russian  title.  A  short  account  of  this  method  will  be  found  in :  Taf el  der 
Klassenanzahlen  fur  Kubische  Zahlkorper,  by  the  author. 


THE    GENERAL   QUADRATIC   REALM.  289 

Thus  in  k(i)>  d[i],  =  —  4,  is  divisible  by  22,  but  I,  its  a  basis 
oik(i). 

§ 6.     Determination  of  a  Basis  of  k(\/m). 

We  have  seen  that  every  number  of  k(^/m)  can  be  written 
in  the  form 

aWi  +  fyv/w, 

where  rt  and  r2  are  rational  numbers. 

Let  rx  =  a/c,  and  r2  =  b/c, 

where  c  is  the  least  common  multiple  of  the  denominators  of  rx 
and  r2,  rx  and  r2  being  in  their  lowest  terms. 

Then  a  = ■,  1) 

c  ' 

where  a,  b  and  c  are  rational  integers  having  no  common  factor. 
The  necessary  and  sufficient  condition  that  a  shall  be  an  integer 
of  k(^/m)  is  that  it  satisfy  an  equation  of  the  form 

x2  -f-  px  +  q  =  o,  2) 

where  p  and  q  are  rational  integers,  the  other  root  of  2)  being 
the  conjugate  of  a;  that  is, 


a'  = 


a  —  bym 


Hence  we  have  as  the  necessary  and  sufficient  conditions  that  a 
shall  be  an  integer  of  k(^m) 

a  +  a'  =  —  a  a  rational  integer,  3) 

#2  —  mb2  «  .  x 

aar  = 2 —  =  a  rational  integer.  4) 

Remembering  that  a,  b  and  c  have  no  common  factor,  and  m  no 
square  factor,  we  shall  show  that  c  can  have  a  value  different 
from  1  only  when  m  =  1,  mod  4,  and  then  can  take  only  the  value 
1  or  2. 
'9 


29O  THE    GENERAL   QUADRATIC    REALM. 

i.  Let  c  =  pc1}  p  being  a  prime  different  from  2.  Then  from 
3)  it  follows  that  a  =  o,  mod  p, 

and  from  4)  that         a2 —  mb2  =  o,  mod  p2, 
and  hence  mb2  =  o,  mod  p2.  5) 

But  5)  is  impossible  since  m  can  not  contain  the  squared  factor 
p2,  and  if  b  were  divisible  by  p  then  a,  b  and  c  would  have  a  com- 
mon factor  p.  Hence  c  can  contain  no  prime  factor  different 
from  2.  s 

ii.  Let  c  =  2e.  We  can  prove  exactly  as  in  i  that  e  can  not  be 
greater  than  1. 

Let  e—  1 ;  that  is,  c  =  2.     Then  from  4)  it  follows  that 

a2  —  mb2  =  o,  mod  4,  6) 

From  6)  we  see  that  a  can  not  be  even,  for  this  would  require 

a2  =  o,  mod  4, 

and  hence  m&2  =  o,  mod  4, 

from  which  it  would  follow  that  either  m  contains  the  squared 
factor  22,  or  a,  b  and  c  have  the  common  factor  2. 

Hence  a  =  2ax  +  *• 

Likewise  b  =  2b  x  +  1 ; 

for  b  even  gives  &2  =  o,  mod  4, 

and  hence  from  4)  a2==o,  mod  4, 

which  we  have  seen  to  be  impossible.  We  see  therefore  that,  if 
c  =  2,  a  and  b  must  both  be  odd  in  order  that  a  may  be  an 
integer;  that  is, 

a  =  2a1-\-i  and  b  =  2b±-\-  1. 

We  must  now  determine  the  form  that  m  must  have  in  order 
that  a2  —  mb2  may  be  divisible  by  4;  that  is,  that  c  may  be  2. 
From  a  =  2ax  -f-  1  and  b  =  2b \  +  I  it  follows  that 

a2  =  1,  mod  4, 

and  &2==  1,  mod  4, 


THE   GENERAL   QUADRATIC   REALM.  29  I 

and  hence  from  a2 —  mb2z==o,  mod  4,  it  follows  that 

1 — m  =  o,  mod  4.  7) 

Therefore  a  and  b  odd  and  w=  1,  mod  4,  are  the  necessary  and 
sufficient  conditions  that  a2  —  mb2  may  be  divisible  by  4.  We 
can  have  therefore  c  =  2  when  and  only  when  these  conditions 
are  satisfied.  Hence,  when  m=i,  mod  4,  every  integer,  a,  of 
&(Vra)  has  the  form 

a  -f-  b\/m 

a — * 

where  a  and  &  are  both  odd  or  both  even,  and  every  number  of 
this  form  is  an  integer  of  &(\/w). 

When  w  =  2or3,  mod  4,  the  condition  7)  not  being  satisfied,  c 
can  not  equal  2,  and  every  integer  of  k(^/m)  has  the  form 

a  =  a-{-  bym, 

where  a  and  b  are  rational  integers.  Every  number  of  this  form 
is  evidently  an  integer  of  k{^/m).  The  cases  m=l,  2  or  3, 
mod  4,  include  all  possible  forms  of  m,  ra  =  o,  mod  4,  being 
excluded,  since  m  would  then  contain  a  squared  factor.  These 
three  cases  are  illustrated  respectively  by  the  realms  &(V — 3), 
&(V2)  and  k(y^i). 

We  shall  now  show  that,  if  w  represent  s/m,  V ra  or  ( 1  +  ■\/m)/2f 
according  as  m  =  3,  2  or  1,  mod  4,  then  all  integers  of  k(^/m) 
can  be  expressed  in  the  form 

a  =  u  +  V(o, 

where  u  and  v  are  rational  integers.  This  is  at  once  evident 
when  w  =  3or2,  mod  4. 

To  show  it  when  w=  1,  mod  4,  we  observe  first  that 

1  -f  Vm 
co  =  — 

2 

is  then  an  integer,  for  it  is  of  the  form  {a-\-byJm)/2,  where  a 
and  b  are  both  odd. 

a  +  bVnt 


Then,  if  a.  = 


292  THE   GENERAL   QUADRATIC    REALM. 

be  any  integer  of  k(^m)  (ra=i,  mod 4),  we  have,  since 

■\/m  =  2(t> —  1, 

a  -f  b(2(o—  1)      a  —  b 

a  = ^ = f-  baa  ; 

2  2 

that  is  a  =  u  +  Vm, 

where  u=(a — b)/2,  v=b  are  rational  integers;  for  a  and  b 
are  rational  integers,  and  (a  —  b)/2  is  an  integer,  since  a  and  b 
are  both  odd  or  both  even. 

Examples. 

1.  Give  a  basis  of  each  of  the  following  realms  :  &(V5),&(\/6), 
k(\/— ¥),  fc(V— "13),  &(V75)  and  £( V^i). 

2.  Tell  whether  each  of  the  following  pairs  of  numbers  is  a 
basis  of  the  realm  to  which  it  belongs,  2  -|-  3  \/6, 1  -j-  -y/6;  1  -+-  V°\ 
7  +  6V6;  K3  +  7V5),  i(-i  — 3VS). 


CHAPTER   XI. 
The  Ideals  of  a  Quadratic  Realm. 

§  i.    Definition.    Numbers  of  an  Ideal. 

An  ideal  of  a  number  realm  is  a  system  of  integers,  alt  aoy  a3, 
•••,  of  the  realm  infinite  in  number  and  such  that  every  linear 

combination,  A^  +  A2a2  +  A3a3  -| ,of  them,  where  Ax,  A2,  A3,  •  •  • 

are  any  integers  of  the  realm,  is  an  integer  of  the  system.1 

The  integers  of  the  infinite  system  which  constitutes  the  ideal 
are  called  the  numbers  of  the  ideal. 

§  2.  Basis  of  an  Ideal.  Canonical  Basis.  Principal  and  Non- 
Principal  Ideals. 

Theorem  i.  There  exist  in  every  ideal  a  of  a  quadratic  realm 
two  numbers,  ix,  i2,  such  that  every  number  of  the  ideal  can  be 
expressed  in  the  form 

i  =  llLl  -|_  /2t2, 

where  lx  and  l2  are  rational  integers. 

Suppose  all  numbers  of  a  to  be  written  in  the  form 

t  =  a1o)1  -f-  a2o)2, 

where  <olf  w2  is  a  basis  of  the  realm,  and  consider  those  for  which 

a2=f=o. 

Among  them  must  be  some  in  which  a2  is  smaller  in  absolute 
value  than  in  any  of  the  remaining  ones. 

Let  t2,  =  b(*x  -[-  c<»2>  De  one  of  these ;  then  c  will  be  the  greatest 
common  divisor  of  the  values  of  a2  in  all  the  numbers  of  a  (see 
Chap.  X,  Th.  i). 

We  have  a2  =  l2c, 

xThe  definition  given  in  &(V^~5)  wiU  be  seen  later  to  coincide  with 
this.     See  also  Hilbert :  Bericht,  p.  182. 

293 


294  THE  IDEALS  OF  A  QUADRATIC  REALM. 

where  l2  is  a  rational  integer,  and  hence 

i  —  12l2  =  (ax  —  l2b  )  m%' 

Consider  now  those  numbers  of  a  in  which  a2  —  o,  but  at=f=o. 
Just  as  before  we  can  show  that  there  exists  among  them  cer- 
tainly one,  i1  =  ao)1,  such  that  a  is  the  greatest  common  divisor 
of  the  values  of  ax  in  all  the  numbers  of  the  ideal  for  which 

a2  =  o,     ax  =4=  o. 
Hence  ax  —  l2b  =  lxa, 

where  lx  is  a  rational  integer,  and 
we  have  i  —  l2h=:hli} 

that  is  i  =  lxix  -\-  12l2, 

hence  ix,  i2  are  the  desired  numbers. 

Any  pair  of  numbers  of  a  such  as  ij,  i2,  having  the  property 
required  by  the  theorem,  is  called  a  basis  of  the  ideal  a.  The  nec- 
essary and  sufficient  condition  that  any  other  pair  of  numbers  of  a 

h*  =  aih  +  a2^ 
t2*  =  bxix  +  b2t2, 

shall  be  a  basis  of  a  is  that 


K   K 


dz  I 


This  condition  can  be  satisfied  by  an  infinite  number  of  sets  of 
rational  integers,  alf  a2,  blt  b2,  and  hence  each  ideal  has  an  infinite 
number  of  bases.  We  shall  call  the  particular  basis  a<aXi  b<ax  +  co>2 
defined  as  above  a  canonical  basis.  Taking  i,«  as  a  basis  of  the 
realm,  we  have  as  a  basis  of  c^a,  b  +  c<o,  an  especially  convenient 
form,  in  which  a  is  evidently  the  rational  integer  smallest  in  abso- 
lute value  occurring  in  a. 

Cor.  i.  //  ax(ox  -J-  a2w2,  bxo)x  -\-  b2<a2  and  cx<ax  -\-  c2o)2,  dxmx  +  d2w2 
be  bases  of  the  same  ideal,  then 


bi     b2\ 


di    d2 


1  See  Chap.  V,  Th.  i. 


THE    IDEALS   OF   A    QUADRATIC   REALM.  295 

Cor.  2.    //  a1o)1  +  a2w2,  b1o)1  -f-  b2<o2  be  a  basis  of  an  ideal,  a, 
and  c1o)1  -f-  c2<a2,  d1w1  -j-  d2<o2  be  any  two  numbers  of  a,  and 


dx    d2 


K     K 


then  cx<ax  -\-  c2to2,  d1(a1  -j-  d2<a2  is  also  a  basis  of  a. 

Th.  1  shows  at  once  that  all  ideals  of  a  quadratic  realm  would  be 
obtained,  if  we  paired  the  integers  of  the  realm  in  all  possible  ways  and 
took  each  pair  a,  j3,  as  defining  an  ideal  (a,  /3)  ;  for  among  these  pairs 
would  be  certainly  a  basis  of  every  ideal  of  the  realm.  In  this  pairing, 
however,  each  ideal  would  be  repeated  an  infinite  number  of  times. 
The  definition  given  of  an  ideal  (§  1)  holds  for  realms  of  any  degree, 
as  does  a  theorem  similar  to  Th.  1 :  namely,  in  every  ideal  of  a  realm  of 
the  nth  degree  there  exist  n  integers,  H,  h,  •  ••,%  such  that  every  number 
of  the  ideal  can  be  expressed  in  the  form  hii  -f-  &*«  +  •  •  •  +  l^n,  where 
h,  U,  •••,  In  are  rational  integers.      See  Hilbert:  Bericht,  Satz  6. 

If  alt  a2,  •  •  •,  ar  be  r  numbers  of  a  such  that  every  number  of  a 
can  be  represented  in  the  form 

\xa2  +  A2a2  +  •  •  •  +  XrCLr,  I  ) 

where  a^  A2,  ---yXr  are  integers  of  the  realm,  we  can  define  a  by 
the  symbol  (a15a2,  --^ar)  ;  that  is,  we  write 

a=  (alya2,  --,ar), 

understanding  thereby  the  infinite  system  of  integers  of  the  form 
1),  the  A's  taking  all  possible  values.  We  shall  call  alfa2f  --^CLr 
the  numbers  defining  the  ideal  a. 

The  numbers  of  a  are  all  those  of  the  form  I ) .  We  may  intro- 
duce into  the  symbol  any  integer  which  is  a  linear  combination  of 
those  already  there  without  changing  the   ideal  defined  by  it. 

Thus,  if  as  =  \tax  +  A2a2  -| 1-  \rCLr, 

we  have        a=  (alfa2f  •••,ar)  =  (a±,a2,  ••-,ar,a,)  ; 
for  the  system  of  integers 

Kai  +  Ka2  H +  ^rCLr 

is  coextensive  with  the  system 

A^i  +  A2a2  +  '  * '  +  ^cir  +  Asa8, 
the  A's  taking  all  possible  values. 


296  THE    IDEALS    OF   A    QUADRATIC    REALM. 

Likewise,  if  any  integer  in  the  symbol  be  a  combination  of  the 
remaining  ones   therein,   it   may   be   omitted    from   the   symbol. 

Thus,  if  ax  =  X2a2  +  A3a3  +  •  •  •  +  Arar, 

we  can  write 

a  =  (alt  a2,  •  •  •,  ar)  =  (a2,  •  •  •, ar) . 

We  speak  for  the  sake  of  brevity  of  (alt  a2,  -  •  • ,  a.r)  as  the 
ideal  a,  and  instead  of  saying  that  we  introduce  a  number,  as, 
into  the  symbol  of  a  or  omit  it  from  the  symbol,  say  that  we 
introduce  a8  into  the  ideal  a  or  omit  it  from  the  ideal,  although  a8 
is  and  remains  a  number  of  a.  It  will  be  convenient  also,  if  ilf  t2 
be  a  basis  of  a,  to  speak  of  (i1?  i2)  as  a  basis  representation  of  a. 
The  determination  of  the  question  whether  an  integer  a  belongs 
to  a  given  ideal  a  will  be  greatly  simplified  by  some  properties  of 
ideals  which  will  be  developed  later.  It  can,  however,  be  easily 
decided  now,  if  we  have  a  basis  of  the  given  ideal,  for  if 
&>  =  ai  +  a2w\  De  any  integer  of  the  realm  and  bx  +  b2u,  ci  +  C2W 
be  a  basis  of  a,  the  necessary  and  sufficient  condition  that  a  shall 
be  a  number  of  a  is  evidently  that  two  rational  integers  lx,  l2 
exist,  which  satisfy  the  equation 

hiPx  +  ^2W)  +  h(ci  ~\~  c2<°)  =^1  +  02w-  2) 

Equating  the  coefficients  of  the  powers  of  o>  in  the  two  mem- 
bers of  2),  we  obtain  the  equations 

b1l1  +  c1l2  =  a1, 

b2l1-\-c2l2  =  a2,  3^ 

which  determine  lx,  l2. 

If  the  values  of  llt  l2  found  from  3)  be  integral,  a  is  a  number 
of  a,  otherwise  not.  If  we  have  not  found  a  basis  of  a,  we  can 
generally  determine  whether  a  is  a  number  of  a  by  means  of  the 
fundamental  condition  that  a  is  or  is  not  a  number  of  a  according 
as  a  is  or  is  not  a  linear  combination  of  the  numbers  defining  a 
with  coefficients  which  are  integers  of  the  realm.  For  an  ex- 
ample of  this  method  see  p.  259. 

1  Unless  the  contrary  be  stated,  1,  w  is  taken  as  a  basis  of  the  realm. 


THE    IDEALS   OF   A    QUADRATIC   REALM.  297 

An  ideal  which  consists  of  all  and  only  those  numbers  of  the 
form  \a,  where  a  is  a  particular  integer  and  A  any  integer  of  the 
realm,  is  .called  a  principal  ideal  and  is  denoted  by  (a).  An 
ideal  not  having  this  property  is  called  a  non-principal  ideal.  For 
examples  of  principal  and  non-principal  ideals  see  Chap.VIII, 
§  10.  It  should  be  observed  that  although  all  numbers  of  a  prin- 
cipal ideal,  (a),  are  multiples  of  the  single  integer  a,  when  as 
multiplier  we  take  any  integer  of  the  realm,  nevertheless,  just  as 
in  the  case  of  a  non-principal  ideal,  a  basis  of  (a)  consists  of 
two  integers,  aoit,  ato2,  where  a^,  <o2  is  a  basis  of  the  realm,  for 
every  number  of  (a)  has  the  form 

{a1(o1  -j-  a2o>2)a  =  axOLu>1  -f-  a2CLo)2) 

where  alf  a2  are  rational  integers. 

For  example:  a  basis  of  (i  -fV — 5)  is  *  +  V — 5>  (i  +V — 5~}x 
V="5 ;  that  is,  i  +  V^5,  —  5  +  V-^T- 

If  the  difference  of  two  integers  a  and  /3  be  a  number  of  the 
ideal  a,  this  fact  is  expressed  symbolically  by  writing 

a  =  ft,  mod  a,  4) 

and  we  say  that  a  is  congruent  to  f3  with  respect  to  the  modulus  a. 
The  fact  that  a  —  (3  is  not  a  number  of  a  is  expressed  symbol- 
ically by  writing 

a 4=/?,  mod  a,  5) 

and  a  is  said  to  be  incongruent  to  /?  with  respect  to  the  modulus 
a.  Every  number,  a,  of  the  ideal  a  is  congruent  to  o  with  respect 
to  the  modulus  a,  or  in  symbols 

a  =  p,  mod  a.  6) 

No  meaning  other  than  the  symbolic  expression  of  the  facts 
mentioned  must  be  attached  for  the  present  to  4),  5)  and  6). 
Thus  we  write 

3  —  2V— I^i  +  sV^,  mod  (7,  S+V^), 
since         3  —  2\/^5—  i1  +2V— ~5)  =2  —  4V—  5 
is  a  number  of  (7,  3  +V — 5),  and  we  write 

i  +  5V^5  +  2  —  3V—  5,  mod(i+2V=r5), 


298  THE  IDEALS  OF  A  QUADRATIC  REALM. 


since        1  +  5  V—  5  —  (2  —  3V~  5)  =—  *  +  8V~  5 
is  not  a  number  of  (1  +  2 V — 5)- 

Although  the  actual  determination  of  a  basis  of  any  given  ideal 
of  a  quadratic  realm  must  be  postponed  until  the  properties  of 
ideals  have  been  more  fully  investigated,  we  can,  however,  now 
determine  whether  any  two  given  numbers  of  an  ideal  a  are  a 
basis  of  a. 

The  necessary  and  sufficient  condition  for  alt  a2  to  be  a  basis 
of  the  ideal  a,=  (alf  a2,  •••,  ar),  is  evidently,  since  every  num- 
ber of  a  has  the  form  A^  +  A2a2  +  •  •  •  +  Arar,  that  for  every 
possible  choice  of  the  A's  we  shall  be  able  to  find  two  rational 
integers,  Ilt  l2,  such  that 

Ai^i  +  A2a2  -\ f-  XrOLr  =  tfo  +  l2CL2.  j) 

Let  w1}  a),  be  a  basis  of  the  realm,  and 
&i  =  ciiOi1  -\-  biO)2  "1 

\i  =  CiWi  -f-  C?iW2  J 

We  have  on  equating  the  coefficients  of  the  number  defining 
the  realm  in  the  two  members  of  7)  two  equations  between 
rational  integers,  whose  satisfaction  by  suitably  chosen  rational 
integral  values  of  lXJ  l2  for  all  possible  choices  of  the  c's  and  d's 
is  the  necessary  and  sufficient  condition  that  alt  a2  shall  be  a 
basis  of  Q. 

*  Ex.  1.  That  3,  1  +  V  —  5  is  a  basis  of  (3,  1  +  V  —  5)  may  be  easily- 
shown  by  the  above  method.  Every  number  of  (3,  1  +  V  —  5)  has  the 
form 

(c1  +  rf1V^5)3+(c2  +  ^V^r5)(i  +  V^5),  8) 

where  Ci,  di,  c«,  d2  are  rational  integers. 

If  3,  i  +  V  —  5  be  a  basis  of  (3,  i  +  V  —  5),  then  every  number  of 
the  form  8)  must  be  expressible  in  the  form  hs  +  /2(i  +  V  —  5)>  where 
h,  U,  are  rational  integers,  and  hence  for  every  possible  choice  of 
Ci,  du  c2,  d2,  we  must  be  able  to  find  rational  integral  values  of  h,  U,  which 
satisfy  the  equation 

(ft  +  <W^5)3  +  ic,  +  hyp-*)  (1  +  V— S)  =  h3  +  4(1  +  V=l), 

or 

3ft  +  cz  —  $d2  -f-  (3di  +  C2  +  d2)  V^  =  34  +  h  +  hy/^J.  9) 


THE    IDEALS    OF   A   QUADRATIC   REALM.  299 

Equating  the  coefficients  of  the  different  powers  of  V  —  5,  we  have 

$Ci  +  c2  —  5^2  =  3/1  +  U  10) 

3di  +  c2  +  d2  =  U,  11) 

as  the  two  equations  whose  satisfaction  by  rational  integral  values  of 
It,  h  for  every  possible  choice  of  Ci,  di,  c2,  d2  is  the  necessary  and  sufficient 
condition  that  3,  1  +  V —  5  shall  be  a  basis  of  (3,  1  +  V  —  S)-  Sub- 
tracting 11)   from  10),  we  obtain 

3*i  —  3di  —  6d2  =  3/1, 

12) 

Zdi  -\-c2-\-d2  =  U, 

a  system  equivalent  to  10),  11),  and  which  evidently  fulfils  the  required 
conditions. 

Hence  3,  1  +  V  —  5  is  a  basis  of  (3,  1  +  V  —  5)-  In  this  particular 
case,  we  might  have  arrived  at  the  result  by  simply  observing  that 
1  +  V —  5  must  be  the  required  basis  number  b  +  cyj  —  5,  since  c  has 
in  1  +  V  —  5  the  smallest  possible  value;  that  is,  1. 

Moreover  3  must  be  the  basis  number  a,  for  if  (3,  1  +  V  —  5)  contain 
a  rational  integer  smaller  in  absolute  value  than  3,  it  would  contain  1 
and  we  should  have 

(3,  i  +  V"=r5)  =  d), 
that  is  easily  shown  to  be  impossible,  the  equation 

(ft  +  ca/^5)3  +  (di  +  <W^~5)  (1  +  V^)  =  1 
not  being  satisfied  by  rational  integral  values  of  Ci,  c2,  di,  d2.    Therefore 
3,  i  +  V  —  5  is  a  canonical  basis  of  (3,  1  + V  —  5). 

Having  shown  that  3,  1  +  V  —  5  is  a  basis  of  (3,  1  +  V  —  5)>  we 
know  that  the  necessary  and  sufficient  condition  for  any  two  numbers, 
S  t2,  to  be  a  basis  of  (3,  1  -f-  V — 5)  is  that 

where  <h$  a2,  bi,  b2  are  rational  integers  satisfying  the  condition 

|*i    **| 

==±i. 

Pi    K 1 

This  condition  is  evidently  satisfied  by  an  infinite  number  of  sets  of  values 
of  (h,  a2,  bu  b2,  from  which  we  obtain  by  13)  an  infinite  number  of 
different  bases  of  (3,  1  -f- V  — 5)-    Thus  since 


II      4 


we  see  that 


3.3  +  1  .  (1 +  V  —  5)  =  io  +  V  —  5 
11.3  +  4(1  +  V— 5)  =37  +  4  V^S 
is  a  basis  of  (3,  1  +  V — 5). 


3oo 


THE    IDEALS    OF    A    QUADRATIC    REALM. 


On  the  other  hand 

ii—  4V^ 
is  not  a  basis  of  (3,  1  +  V 
2 


5  =  2.3  —  i(i  +  V  —  5), 

5=5-3— 4(l  +  \F"S). 

-5),  since 

-  I  I 


=  -3*±i. 
5     -4| 

By  means  of  the  condition  given  in  Th.  1,  Cor.  1,  it  may  be  shown  even 
more  easily  that  5 — V— 5,  11 — 4V — 5  is  not  a  basis  of  (3,  1  +  V—  5)  ; 
for  1,  V  —  5  being  a  basis  of  the  realm,  we  have 

3  =  3-1+0-  y/~=5,        5  —  V^5  =  5  •  I  +  —  I  •  V7^, 

1  + V^5=I  •  1  +  1  •  V"37!     11— 4VT775  =  ii  •  i+  — 4-  V^, 


5 


11     -4 


Ex.  2.     We  can  show  in  like  manner  that 


■3  +  V 


3t   3  +  5V  — 3 
2 


is  not  a  basis  of  the  ideal  (— 2 +  »,'  — 1 +  5&O  of  the  realm  &(V  —  3). 

1,    *  ~r  V       3  being  taken  as  a  basis  of  the  realm. 
2  0 

Proceeding  as  in  Ex.  1  we  see  that  the  necessary  and  sufficient  con- 
dition for  —  2  +  «,  —  1  4-  5W  to  be  a  basis  of  the  given  ideal  is  that  the 
equation 

(d  +  <*»*)  (—  2  -f-  «)  -f-  (c2  +  &«)  (_  1  4.  5co) 

14) 

=  /1(-2  +  o,)+/2(-I  +  5«) 

shall  be  satisfied  by  rational  integral  values  of  h,  h  for  every  possible 
choice  of  cu  di,  c2,  d2. 

Performing  the  multiplications  indicated  in  14),  putting  w"=  —  1  +  w, 
and  equating  coefficients  of  like  powers  of  «  in  the  two  members,  we  have 
the  equations 

—  2Ci  —  C2  -\-  di  —  5rf2  s=  —  2/1  —  h, 


which  give 


These  equations  evidently  do  not  give  integral  values  for  h,  U  for 
every  possible  choice  of  Ci,  di,  c2,  d2 ',  for  example,  for  a  =  dx  =  c2  =  d2  =  1. 
Hence  — 2  -f-  w,  — 1  -f-  5W  is  not  a  basis  of  ( — 2  -)-  w,  — i_[_5o>).  We 
have  chosen  an  ideal  of  the  realm  k{\/ — 3),  in  which  the  unique  fac- 
torization law  holds  in  the  ordinary  sense,  to  emphasize  the  fact  that 
with  the  introduction   of   ideals   all   quadratic   realms   are  to  be  treated 


1  +  5^2 

—  3^i- 

-6d2- 

~-h  +  5k 

—  9^1 

+  2fl?x- 

-31&2 

=  —  9k 

9c2 

—  5^i- 

-  I7c?2  - 

=  9/2, 

THE    IDEALS    OF   A    QUADRATIC    REALM.  30I 

alike,  and  that  all  theorems  to  be  deduced  hereafter  will  be  equally  valid 
whether  the  unique  factorization  law  holds  in  the  ordinary  sense  or  not. 

Ex.  3.  Show  both  by  the  above  method  and  by  the  nature  of  a  canonical 
basis  that  7,  3  +  V~  5  is  a  basis  of  the  ideal  (7,  3  +  V=r5)  ;  also  that 
3  +  V— 5,    5  +  4V  — 5    is   a   basis    of   the    same   ideal.     In    k(yj^23) 

show  that  3,  I  +  >^~23   is  a  basis  of  the  ideal   (3,   *  +  ^~  2A  .  aiso 

q  _1_  ^-v  / 2  "3 

that  4  +  V —  23,  —         is  a  basis  of  the  same  ideal. 

In  k{yJ6)  showthat  10  +  $\/6,  6-\-2\J6  is  a  basis  of  the  ideal 
(10  +  3V6,  6  +  2V6).  

Ex.  4.  Show  that  7  -f  7 V—  5,  —  5  +  3V— 5  is  not  a  basis  of  the  ideal 
(7  +  7V— 5,    — 5  +  3V— 5).      _ 

Ex.  5.     Show    that    (3*  —  J     is    a    principal    ideal    of    &(V~i3)- 

Show  that  the  two  ideals   (2,  I  +  ^~  I5  )    and    (3,  ~  +  ^~  I5  )    are 

both  non-principal  ideals  of  k(yj — 15),  but  that  their  product  is  a  prin- 
cipal _JdeaL      Show    that    (2,    1  +  V — 13)    is    a    non-principal    ideal    of 

*(V— 13). 

§  3.    Conjugate  of  an  Ideal. 

//  a  be  any  ideal,  the  ideal,  zvhose  numbers  are  the  conjugates 
of  the  numbers  of  a,  is  called  the  conjugate  of  a  and  is  denoted 
by  a'.1  It  is  easily  seen  that,  if  a=  (alt  a2,  •  •  •,  a„)  be  any  ideal, 
then  a'  =  (a/,  a/,  •••',  a/)  is  the  conjugate  of  a;  for,  if 

^iai  +  A2a2  +  *  "  '  +  ^nCln 

be  any  number  of  a,  its  conjugate 

A/a/  +  A/a/  H h  A»'a»' 

is  a  number  of  a,  and  vice  versa. 

Moreover,  if  axwx  -f-  a2o>2,  fr^  +  62w2  be  a  basis  of  a,  where 
*ti  o>2  is  a  basis  of  the  realm,  then  a^/  +  «2°)2/>  ^iwi'  +  ^^  is  a 
basis  of  a'.  The  truth  of  the  last  statement  is  readily  seen  when 
we  remember  that,  if  a1(o1  +  a2w2,  fr^  -f-  b2a>2  be  a  basis  of  a, 
then  every  number,  a,  of  a  can  be  expressed  in  the  form 

a==a(a1oi1  -}-a2a>2)  -f-  b{b1(a1  +  &2<o2), 
where  a  and  &  are  rational  integers. 

The  corresponding  number,  a',  of  a',  being  expressible  in  the 

form  a,=  a(a1(i)l'  +  a2o>/)  +  b^b^  +  b2o)2), 

it  is  evident  that  a^/  +  a2w/,  &!<»/  +  &2<o/  is  a  basis  of  a'. 
1Hilbert:  Bericht,  p.  191. 


302  THE    IDEALS    OF   A    QUADRATIC    REALM. 


For  example,  the  conjugate  of  (2  +  3 V — 5,  7  +  2V — 5>  *7) 
is  (2  — 3Vzr5,  7^2y^5,  17)^ also  since  3,  1  +V^5  is  a 
basis  of  (3,  14.y-~.-5),  3,1—  V:=r5  is  a  basis  of  (3>x—  V—  5)- 

§  4.    Equality  of  Ideals. 

Two  ideals,  a,=  (a1,a2,---,ar),  and  &,=  (/?i,/?2>  ••-,/?«),  are 
said  to  be  equal,  and  we  write  a  =  B,  when  every  number  of  a 
is  a  number  of  fc  and  every  number  of  fc  is  a  number  of  a. 

The  necessary  and  sufficient  condition  for  the  equality  of  a 
and  h  is  that  every  number,  ai,  defining  a  shall  be  expressible 
in  the  form 

ai  =  A101  +  A2flH hA8/?s, 

and  that  every  number,  /?y,  defining  b  shall  be  expressible  in  the 
form  pj  =  fijO^  +  fi2a2  -\ f-  {xrar. 

The  practical  test  of  equality  is  to  see  whether  the  symbol 
defining  either  one  of  the  ideals  can  be  reduced  to  that  defining 
the  other  by  the  introduction  and  omission  of  numbers  under  the 
laws  given  in  the  preceding  paragraph.1 


Ex.  1.     Show  that  (6  +  2V  —  5,  56  +  7V~ 5)  =  05  +  5V  —  5,  14). 
Ex.  2.    Show   that 

(I+2>/l3,  5+8/^3,  5  +  2  fft$)  =  (5  +  14  1A3,  6/IE3). 

Ex.  3.    Show  that   (7,  1  +V=i3)  4=  (7,  1— V=i3). 
§  5.    Multiplication  of  Ideals. 
By  the  product  ah  of  the  two  ideals 

w  understood  the  ideal,  whose  numbers  consist  of  all  possible 
products  of  a  number  of  a  by  a  number  of  b,  together  with  all 
linear  combinations  of  such  products  with  coefficients  which  are 
any  integers  of  the  realm.2 

We  have  therefore 

ah  =  (aA,  •  •  •, a^,  •  -,OrA,  •  •  •, arp8), 

'See  Chap.  VIII,  §9. 

2Hilbert:  Bericht,  p.  183;  also  see  Chap.  VIII,  §  11. 


THE    IDEALS   OF   A    QUADRATIC   REALM.  303 

where  the  numbers  defining  qB  are  all  possible  products  of  the 
numbers  defining  a  by  those  defining  b. 

If  a=(a)  and  5=  (ft, ft,  —,ft), 
then  oB  =  (aft, aft,  •  •  •, aft). 

If  a=(a)  and  B=(0), 

then  ab=(a£), 

and  we  see  that  the  product  of  two  principal  ideals  is  a  principal 
ideal. 

It  is  evident  from  the  definition  that 

ah  =  ba, 

and  that  ctb-c  =  a-bc; 

that  is,  that  the  commutative  and  associative  laws  of  multiplica- 
tion hold  for  ideals. 
Ex.  Show  that 

(2,  V^6)  (3,  i  -  V^6)  (5,  2  +  V^26)  =  (2  +  V  -26). 

§  6.    Divisibility  of  Ideals.    The  Unit  Ideal.    Prime  Ideals. 

An  ideal,  a,  is  said  to  be  divisible  by  an  ideal,  b,  when  there 
exists  an  ideal,  c,  such  that 

a  =  bc. 

We  say  that  b  and  c  are  divisors  of  o,  and  that  a  is  a  multiple 
of  b  and  c.  We  have  as  a  direct  consequence  of  the  above 
definition : 

//  each  of  a  series  of  ideals  alt  ct2,  a3,  •••,  be  a  multiple  of  the 
next  following  one,  then  each  is  a  multiple  of  all  that  follow. 

If  two  or  more  ideals,  a,  b,  c,  •••,  be  each  divisible  by  an  ideal 
j,  j  is  said  to  be  a  common  divisor  or  common  factor  of  a,  b,  c,  •  •  •. 

Theorem  2.     //  the  ideal  a  be  divisible  by  the  ideal  b,  then  all 
numbers  of  a  belong  to  b. 
For  suppose  that 

a  =  bc, 


304  THE    IDEALS    OF   A    QUADRATIC    REALM. 

where 

a=(apa2,"-,ar),  b=  (ft,ft,  ••-,/?*),  c=  (yi,y8,  •••,?#); 
then  a=(ftyi,  "-fPxyt,  •■•,fty„  •••jftyt)- 

The  numbers,  fty15  ■•■,(3syt,  defining  a  are  seen  to  be  numbers 
of  ft.     Hence  all  numbers  of  a  are  numbers  of  B. 
Therefore 

h  =  (ft,  ft,  ••-,/?*,  aif  a2,  •  •  •,  ar), 
and  c==  (yi,y2>    "$yu <h><**>  •••,otr). 

Cor.  1.  //  fcco  ideals  fo  .swc/j  fto  £?ac/i  w  a  divisor  of  the 
other,  they  are  identical. 

The  converse  of  Theorem  2  is  also  true ;  that  is,  if  all  numbers 
of  a  be  numbers  of  b,  a  is  divisible  by  h,  but  its  proof  must  be 
deferred  until  some  necessary  theorems  have  been  demonstrated. 

Every  ideal  is  divisible  by  the  ideal  (1),  which  consists  of  all 
integers  of  the  realm.     Therefore   (1)   is  called  the  unit  ideal. 

The  only  ideal  having  this  property  is  evidently  (1),  for  every 
divisor  of  (1)  contains  all  integers  of  the  realm  and  is  (1).  We 
observe  that  (r/)  =  (1),  where  77  is  any  unit  of  the  realm. 

Since  (i)ct=a,  there  is,  in  the  case  of  ideals,  no  distinction  to 
be  made  corresponding  to  that  made  between  associated  integers. 
An  ideal,  not  the  unit  ideal  and  divisible  only  by  itself  and  the 
unit  ideal,  is  called  a  prime  ideal. 

In  k{ V=5),  (2,  1  +V^),  (3,  1  +V::^5),  (3.  1  -V— S) 
were  shown  to  be  prime  ideals  (see  p.  264). 

Two  ideals  are  said  to  be  prime  to  each  other  when  they  have 
no  common  divisor  except  (1).  Two  integers  a  and  fS  of  the 
realm  are  said  to  be  prime  to  each  other  when  the  principal  ideals 
(a)  and  (ft)  are  prime  to  each  other. 

For  the  sake  of  brevity  we  shall  often  say  that  an  integer  a 
is  divisible  by  an  ideal  a,  instead  of  saying  that  the  principal  ideal 
(a)  is  divisible  by  a.  The  latter  meaning  is,  of  course,  always  to 
be  understood.  Similar  expressions,  such  as  "  a  prime  to  a," 
"  the  greatest  common  divisor  of  a  and  a,"  etc.,  are  to  be  taken 
in  the  same  sense. 


THE   IDEALS   OF   A   QUADRATIC   REALM.  305 

By  means  of  the  definition  of  divisibility  and  the  fact  that 
every  ideal  has  a  basis,  we  can  prove  the  following  important 
theorem : 

Theorem  3.  An  ideal  j  is  divisible  by  only  a  finite  number  of 
different  ideals.1 

Let  a  =  (ao)1  +  b<a2,  cw1  +  d<a2) 

be  a  divisor  of  j,  where  aoi1  -f-  bo)2,  co)1  -f  du>2  is  a  basis  of  a,  <olt  <o2 
being  a  basis  of  the  realm. 

Let  /?  be  any  number  of  j.     Then,  since 

n  [/?]=/?/?' =  0,  mod  i, 

and  a  is  a  divisor  of  },  we  see  that  by  Th.  2 

n[fi]  e=o,  mod  a; 

that  is,  the  rational  integer  n[(3]  belongs  to  every  divisor  of  \. 
Denote  now  w[/3]  by  n  and  let  alf  bx,  clt  d1  be  the  smallest  posi- 
tive remainders  of  a,  b,  c,  d  with  respect  to  n.     Then 

a  =  (aio-L  -f-  b(o2,  cuix  -\-  do)2,  n) 

=3  {a1o>1  -f-  bxu)2,  c^-^  -\-  d^^  n)  1) 

Suppose  every  divisor  of  j  to  be  expressed  in  the  form  1). 
Since  alt  blf  cls  dx  can  each  take  only  the  finite  number  of  values 
o,  1,2,  •••,  \n\  —  1,  it  is  evident  that  the  number  of  different 
divisors  of  \  is  finite. 

§  7.    Unique  Factorization  Theorem  for  Ideals. 

We  shall  now  proceed  to  prove  the  theorem  whose  truth  is  the 
raison  d'etre  of  the  ideal;  that  is,  that  every  ideal  can  be  repre- 
sented in  one  and  only  one  way  as  a  product  of  prime  ideals. 

This  theorem  will  enable  us  to  develop  for  the  ideals  of  the 
general  quadratic  realm  a  series  of  theorems  similar  to  those 
already  given  for  the  integers  of  certain  realms  in  which  the 
ordinary  unique  factorization  theorem  held. 

The  proof  of  the  unique  factorization  theorem  for  the  ideals 

1Hilbert:  Bericht,  Hiilfsatz  1. 
20 


306  THE  IDEALS  OF  A  QUADRATIC  REALM. 

of  the  general  quadratic  realm  will  be  very  like  that  for  the 
integers  of  R,  &(V — i),  k(\/ — 3)  and  &(V2)-  It  depends 
directly  upon  the  theorem  that,  if  the  product  of  two  ideals  be 
divisible  by  a  prime  ideal,  at  least  one  of  the  factors  must  be 
divisible  by  this  prime  ideal.  The  latter  theorem  is  a  consequence 
of  a  series  of  three  theorems  which  have  no  analogues  in  those 
relating  to  integers.  It  depends,  in  the  first  place,  directly  upon 
the  theorem  referred  to  on  p.  304,  that,  if  all  the  numbers  of  an 
ideal  belong  to  another  ideal,  the  first  ideal  is  divisible  by  the 
second.  This  depends,  in  turn,  upon  the  theorem,  that,  if  the 
products  ab,  etc  of  two  ideals,  b  and  c,  by  a  third  ideal  a  be  equal, 
then  b  =  c,  and  this  upon  the  theorem,  that  for  every  ideal  there 
exists  another  ideal  such  that  the  product  of  the  two  is  a  principal 
ideal. 

This  last  theorem  is  the  starting  point  of  the  proof  of  the 
unique  factorization  theorem  and  needs  for  its  demonstration  a 
theorem  which  we  shall  proceed  to  give. 

Theorem  4.  //  the  coefficients,  alt  a2,  J31}  /?2  of  the  two  ra- 
tional integral  functions  of  x, 

4>{x)  ==a1x-\-a2  and  \f/(x)  =/3±x  +  /?2, 

be  integers  of  k(\/m)  and  w,  an  integer  of  k(\/m),  divide  each  of 
the  coefficients,  y19  y2,  y3,  of  the  product  of  the  two  functions, 

F  (x)  =  <f>(x)  ^(x)  =  a^x*  +  (fllft  +  a£x)x  +  a2(32 

=  71^  +  72^  +  73, 

then  each  of  the  numbers  a^ly  a^2,  a2f31}  axp2  is  divisible  by  w.1 
Suppose  ax  and  /?!  =|=  o.     Then  y1  =(=  o.     We  have 

Hence  — a2/?i/7i  and  — ^A/Vi  are  tne  roots  of 

1  Hurwitz :  Nachr.  der  K.  Ges.  der  Wiss.  zu  Gottingen,  1895 ;  also  Hil- 
bert:  Bericht,  Hiilfsatz  2. 


THE    IDEALS   OF   A    QUADRATIC   REALM.  307 

Let  £  represent  either  a2px  or  a1fS2 ;  we  have 

V    vj     7i\    rj    % 

and,  multiplying  this  equation  by  yx2, 

€°~  —  y2£  +  7i73  —  o. 

Since  y2  and  yiH  are  divisible  by  w  and  o>2  respectively,  the 
coefficients  of  the  equation 


i-Y-H1) 

\(0  )  to  \  to  J 


+     O)2  °' 


that  £/o>  satisfies,  are  integers.     Hence  £/«  is  an  integer  (Chap. 
IX,  Th.  9,  Cor.  1 )  ;  that  is,  ax/32  and  a^  are  divisible  by  ». 

Theorem  5.  For  ^z/^ry  icfea/  a  of  a  quadratic  realm  there 
exists  an  ideal  h  of  the  realm  such  that  the  product  ah  is  a  prin- 
cipal ideal.1 

Let  a=  (alt  a2)  where  alf  a2  is  a  basis2  of  a.  We  shall  show 
that  the  conjugate  of  a,  that  is,  the  ideal  &,=  (a1',  a2'),  where 
a/,  a2'  are  the  conjugates  of  alf  a2,  has  the  desired  property.3 

Let  <f>(x)=a1x  -{-a2  and  ij/(x)  =  a1,x  +  a2'. 

Form  the  product 

4>(x)iKx)  =  a1a1,x2  -f-  (a^a/  +a1,a2)jir  H-c^A/ 

=  7i^2  +  y2^  +  y3- 

Let  0  be  a  number  defining  the  realm  and  let  the  irreducible 
rational  equation  of  which  6  is  a  root  be 

x2  -\-  arr  -f  a2  =  o.  1 ) 

Since  yu  y2,  y3  are  symmetric  functions  of  the  roots  of  1),  they 

'Hilbert:   Bericht,   Satz  8. 

"  The  simplification  effected  by  the  use  of  the  basis  representation  of  an 
ideal  is  that,  in  a  quadratic  realm,  the  basis  consists  of  two  numbers  and 
hence  Th.  4  need  be  proved  only  for  functions  of  the  first  degree. 

3  In  the  realm  of  the  nth  degree  the  ideal  that  will  have  the  desired 
property  is  the  product  of  the  conjugates  of  a.  This  ideal  is,  however, 
not  the  only  ideal  having  the  desired  property  (Chap.  XIV,  §  1). 


3<d8  the  ideals  of  a  quadratic  realm. 

are  rational  integral  functions  of  its  coefficients  i,  alt  a2.  Hence 
Yi>  72j  73  are  rational  numbers.  But  ylt  y2,  y3  are  also  integers, 
since  alf  a/,  a2,  a2  are  integers  (Chap.  IX,  Th.  8,  Cor.  2). 
Hence  yu  y2,  y3  are  rational  integers. 

Let  a  be  the  greatest  common  divisor  of  ylf  y2,  y3.     Then 

ah=(a1a1',  a,a2',  atfa2,  a2a2') 

is  equal  to  the  principal  ideal  (a)  ;  for  by  Chap.  II,  Th.  B,  we 
can  find  three  rational  integers,  fw  t2,  t3,  such  that 

a  =  ^i7i  +  *2y2  +  hys 
=  f^a/  +  ^(o^a,/  +  a/a2)  +  tza2a2' 

Hence  a  is  a  number  of  ah  and  we  have 

ah  =  (a1a1't  axa2,  a^a2,  a2a2,  a). 

But  by  Th.  4  each  of  the  numbers  a^',  a±a2,  a^a2,  a2a2  is  a 
multiple  of  a.     Hence  we  can  omit  them  from  the  symbol  and 

have  ah=(a). 

Therefore  b  is  the  required  ideal. 

It  will  be  observed  that  we  have  proved  that  the  product  of  an  ideal 
of  a  quadratic  realm  by  its  conjugate  is  a  rational  principal  ideal.  This 
will  be  of  use  later. 

Theorem  6.     //  a,  h  and  c  be  ideals  and  ac  =  6c,  then  a^b.1 
Let 

a=(a1,a2,--,ar),  h=  (ft, ft,  ••-,&),  c=  (yi,y2,  •••,yt), 

and  let  m,  =  (filt  fi2,  --,iin),  be  an  ideal  such  that 

cm=(y1^1,  -,y</in)  =  (a), 
a  principal  ideal. 
Then  ocm  =  km, 

or  a(a)=h(a), 

or  (axa, a2a,  •  •  •,  ara)  =(fta, fta,  ■  •  •,  fta). 

Since  these  two  ideals  are  equal,  every  number  of  the  one  must 

1  Hilbert :  Bericht,  Satz  9. 


THE    IDEALS   OF   A   QUADRATIC   REALM.  309 

be  a  linear  combination  of  the  numbers  defining  the  other,  with 
coefficients  which  are  integers  of  the  realm. 

Hence,  if  aia  be  any  number  of  the  first  and  PjO.  any  number 
of  the  second,  we  have 

aid  =  ^pxa  +  £2p2a  +  •  •  •  +  ZsPsCL, 

and  pjOL  =  rj^OL^  -f~  f]2CL20.  +  •  •  •  -f-  TfrCLfOL, 

where  the  £'s  and  rfs  are  integers  of  the  realm.     Hence 

a«=!a&  +  *AH \-i*fr, 

Pj  =  n^  +  r)2a2  -j [-  ^a,.. 

Hence  every  number  of  a  is  a  number  of  b,  and  every  number 
of  h  is  a  number  of  a,  and  consequently 

a  =  6. 

Theorem  7.     7/  a//  numbers  of  an  ideal  c  belong  to  an  ideal 
a,  c  is  divisible  by  a.1 

Let        c=  (yu  ■■-,yt)  and  a=  (alf  •••,ar,y1,  •••,y*)  ; 

and  let  m,=  (f^,  ••-,/*«),  be  an  ideal  such  that 

ftm=(o1/H,  •  •  • , ar^n, yi/*!,  •••,y*/*«)  =  (a), 

a  principal  ideal. 

Then  all  numbers  of  am,  and  hence  Yifi1,'",yil**,'"tytiHf'"t 
ytfin,  must  be  divisible  by  a.     Hence  all  numbers  of 

rcic=  (yi/^i,  •••>yiMn,  •••>y*A*n) 
are  divisible  by  a ;  that  is, 

mc  =  (vxa,  •  •  • ,  vn*a)  =  (a)  (vx,  •  •  ■ ,  v„« )  =  (a)b.  2) 

Multiplying  both  members  of  2)  by  a,  we  have 
cmtc=  (a)o6, 
or  c  =  ah. 

Hence  c  is  divisible  by  a. 
1Hilbert:  Bericht,  Satz  10. 


3IO  THE  IDEALS  OF  A  QUADRATIC  REALM. 

This  theorem  justifies  our  use  of  the  notation 

a  =  o,  mod  a, 

to  denote  that  (a)  is  a  multiple  of  a.  For,  if  a  be  a  number  of  a, 
then  from  the  above  theorem  it  follows  that  (a)  is  divisible  by  a. 
From  Th.  2  we  saw  that  a  necessary  condition  for  an  ideal  a 
to  be  divisible  by  an  ideal  b  is  that  all  numbers  of  a  shall  belong 
to  b;  from  Th.  7  we  see  that  this  condition  is  also  sufficient. 
Hence  every  common  divisor,  b,=  (8lf  •■■,8t),  of  two  ideals 

a=(a1,  •••,Or),    &=(&,  •••,£,) 

must  contain  all  numbers  of  both  a  and  b ;  that  is, 

where  8X,  •••,$*  are  any  integers  of  the  realm,  and  every  ideal  of 
this  form  is  a  common  divisor  of  a  and  b. 

Among  the  common  divisors  of  a  and  b  is  one,  g,  to  which 
belong  no  numbers  other  than  the  numbers  of  a  and  b,  together 
with  the  linear  combinations  of  these  numbers;  that  is, 

This  ideal  g  is  divisible  by  every  common  divisor,  b,  of  a  and 
b,  for  b  must  contain  all  numbers  of  a  and  b,  and  hence  all  the 
numbers  of  g,  and  therefore  is  a  divisor  of  g. 

As  in  the  case  of  rational  integers,  g  is  called  the  greatest 
common  divisor  of  a  and  b. 

That  g  is  the  only  ideal  having  this  property  is  evident ;  for  did 
a  second,  f),  exist,  then  g  must  be  divisible  by  I)  and  I)  by  g,  and 
hence  g  and  §  be  identical  (Th.  2,  Cor.). 

Likewise  the  necessary  and  sufficient  condition  that  an  ideal,  tn, 
shall  be  a  common  multiple  of  a  and  b  is  that  all  numbers  of  m 
shall  be  common  to  both  a  and  b. 

Among  the  common  multiples  of  a  and  b  is  one  to  which  belong 
all  numbers  common  to  both  a  and  b,  together  with  the  linear 
combinations  of  these  numbers. 

This  ideal,  I,  is  evidently  a  divisor  of  every  common  multiple 


THE  IDEALS  OF  A  QUADRATIC  REALM.  3  1  I 

of  a  and  b.  That  I,  moreover,  is  the  only  ideal  having  this  prop- 
erty may  be  shown  as  in  the  case  of  g. 

As  in  the  case  of  rational  integers,  I  is  called  the  least  common 
multiple  of  a  and  b. 

We  shall  denote  the  greatest  common  divisor  of  a  and  b  by 
the  symbol  a  +  b,  and  the  least  common  multiple  of  a  and  b  by 
the  symbol  a — b.  No  idea  of  addition  or  subtraction  is  to  be 
conveyed  by  these  symbols. 

From  Theorems  2  and  7  we  have  the  important  result  that  an 
ideal  a,=  (a1,Oa,  •••,ar),  is  the  greatest  common  divisor  of  the 
numbers  defining  it  considered  as  principal  ideals;  that  is,  a  is 
the  greatest  common  divisor  of  (o^),  (a2),  •••,  (ar). 

The  fact  that  we  can  at  once  write  the  greatest  common  divisor 
of  any  number  of  ideals  by  placing  in  a  single  symbol  all  the 
numbers  defining  the  ideals  is  of  use  in  numerical  work  with 
ideals.  Thus,  if  we  can  show  that  the  greatest  common  divisor 
of  two  ideals  so  written  is  (1),  we  know  that  the  ideals  are  prime 
to  each  other. 

Ex.  The  greatest  common  divisor  of  (3 -f- V  —  5)  and  (8-f-V  —  5) 
is  (3  +  V  —  5>  8  +  V  —  5),  and  we  have 

(3  +  V^5,  8  +  V^l)  =  (3  +  V"=r5,  8+V=5,  5,   14) 

=  (3  +  V=:5,  8  +  V=I),  5,   14,   0  =  0) 
Hence  (3  -+-  V  —  5)  and  (8+V  —  5)  are  prime  to  each  other. 

The  ideas  of  the  greatest  common  divisor  and  least  common 
multiple  of  two  ideals  may  be  at  once  extended  to  any  number  of 
ideals. 

Thus,  if  alt  a2,  •  •  •,  am  be  any  number  of  ideals  of  a  realm,  there 
is  among  the  common  divisors  of  a^cio,  ■•■,am  one,  g,  to  which 
belong  no  numbers  other  than  the  numbers  of  alf  a2,  •••,  Qm, 
together  with  the  linear  combinations  of  these  numbers;  that  is, 

if      a1=(a1,-'-,ar),  0*=  (ft, •••,/*•  ),•••,  fc»=G«i> •••»**#), 

then  g=  (alt  •••,ar,^1,  ••-,/?*,  ••-,/*!,  ••-,!»#). 

That  g  is  divisible  by  every  common  divisor  of  av  a2,  •••,am 
and  is  the  only  ideal  having  this  property  is  seen  as  in  the  case  of 
two  ideals.     We  call  g  the  greatest  common  divisor  of  alf  a2,  •  •  • ,  am . 


312  THE  IDEALS  OF  A  QUADRATIC  REALM. 

Likewise  the  ideal,  I,  to  which  belong  all  numbers  common  to 
a19  a2,  --'yOm,  together  with  their  linear  combinations  and  no 
others,  is  evidently  the  only  common  multiple  of  a:,  a2,  •  •  •,  am  that 
is  a  divisor  of  every  common  multiple  of  alf  a2,  •••,  am.  It  is 
therefore  called  the  least  common  multiple  of  alfa2i  •••,om. 

We  write  symbolically 

Q  =  a1  +  a2-\ \-0mt 

and  l  =  a±  —  a2 — ••• — dm- 

We  have  as  an  immediate  consequence  of  Th.  7  and  the  defini- 
tion of  the  least  common  multiple  of  two  or  more  ideals  the 
following : 

Cor.  //  an  ideal  a  be  divisible  by  each  of  the  ideals  hlt  b2,  •••, 
Br,  then  a  is  divisible  by  the  least  common  multiple  of  ht,  b2,  ■  ••,  br. 

We  shall  see  later  that  the  greatest  common  divisor,  as  defined 
above  for  ideals,  possesses  the  remaining  two  properties  which 
distinguished  the  greatest  common  divisor  of  two  or  more  integers 
in  those  realms  in  which  the  unique  factorization  law  held  in  the 
ordinary  sense  (see  p.  318). 

We    have    now    a    full    justification    of    our    introduction    in 

feCV31^)  of  the  ideals  (2^_! +Vzr5),  (3,  i+V^),  (2> 
1 — V — 5)  and  (3,  1  —  V — 5)  as  the  greatest  common  divisors 
respectively  of  (2)  and  (i+V — 5)»  (3)  anc*  (1  +V — 5),  (2) 
and  (1— Vzr5),and  (3)  and  (i— -yCTj)- 

Th.  7  having  been  proved,  the  remaining  theorems  necessary 
for  the  proof  of  the  unique  factorization  theorem  and  the  proof 
of  that  theorem  itself  for  ideals  are  strictly  analogous  to  the  cor- 
responding theorems  in  the  realms  in  which  the  unique  factoriza- 
tion law  held  in  the  ordinary  sense. 

It  may  seem  singular  that  the  divisors  of  an  ideal,  a,  are  in  a  way 
larger  systems  of  numbers  than  the  ideal,  a,  itself;  that  is,  they  contain 
not  only  the  numbers  of  a  but  in  addition  any  other  numbers  of  the 
realm  that  we  choose  to  introduce. 

When,  however,  we  remember  that  by  Th.  7  an  ideal  divides  every  one 
of   its   numbers   considered   as    a  principal   ideal,   it   is   evident   that,   in 


THE    IDEALS   OF   A   QUADRATIC   REALM.  313 

general,  the  more  numbers  we  introduce  into  the  symbol  of  an  ideal, 
that  are  not  linear  combinations  of  those  already  there,  so  much  the  more 
do  we  narrow  the  ideal  by  thus  placing  more  restrictions  upon  it. 

For  example;  the  ideal  (14,  3 -f  V  — 14)  is  the  greatest  common  di- 
visor of  (14)  and  (3  +  V — 14)»  and  the  ideal  (14,  3  +  yJ~ZT^  2), 
that  contains  all  numbers  of  (14,  3  +  V  — 14)  and  other  numbers  be- 
sides, divides  not  only  (14)  and  (3 -f- >/ -^4),  and  hence  is  a  divisor 
of   (14,  3  +  V  —  J4)»  DUt  must  also  divide  (2). 

It  is  analogous  to  the  case  of  rational  integers  when  we  observe  that  120 
is  divisible  by  every  common  divisor  of  120  and  18,  and  that  every  common 
divisor  of  120  and  18  is  divisible  by  the  common  divisors  of  120,  18  and  4. 

Examples. 

1.  Find  the  greatest  common  divisor  of  (8  +  V — 14)  and 
(4— V— 14). 

2.  Find  the  greatest  common  divisor  of  (26,  10-J-2V —  14, 
13 V—  14,  —  14+5 V— 14)  and  (10,  2  +  2V— 14,  5V— 14, 
—  14+V— 14).  

3.  Show  that  the  two  ideals  (5,  — 4+V-—14)  and  (13, 
5  —  12  V —  14)  are  prime  to  each  other. 

4.  Making  use  of  form  of  canonical  basis,  show  that  (23, 
8  -f-  V —  5)  is  a  prime  ideal. 

5.  Show  that  (p,  b  -\-o))  is  a  prime  ideal,  p  being  a  rational 
prime,  b  any  rational  integer,  and  1,  w  a  basis  of  the  realm. 

6.  If  p  and  q  be  two  different  rational  primes,  show  that  in  no 
realm  can  (p)  and  (q)  have  a  common  ideal  factor  different 
from  (1). 

7.  Show  that  (1  +V — 5)  is  tne  least  common  multiple  of 
(3,  1  +V=S)  and  (2,  1  -f-V^)-  

8.  Find  the  least  common  multiple  of  (6,  4+V — 14)  and 
(10,  6+V— 14)- 

9.  Show  that,  if  a  be  divisible  by  ax  and  6  by  6X,  then  ab  is 
divisible  by  a^. 

10.  Show  that,  if  ah  be  divisible  by  QC,  then  b  is  divisible  by  c 
and  in  particular  that,  if  a  be  divisible  by  ab,  then  b=  (1). 

11.  Show  that,  if  a,  B  and  c  be  any  ideals,  then 

[a  +  B]c=oc  +  Bc. 


314  THE    IDEALS    OF   A   QUADRATIC    REALM. 

12.  Show  that 

[a  +  B  +  c]  [Bc  +  ca  +  aB]  =  [B  +  c][c  +  a][a  +  B]. 

13.  Show  that,  if  a  be  divisible  by  ax,  and  ft  by  hx,  then  a  +  B 
is  divisible  by  ax  +  Bi,  and  also  that  a  —  B  is  divisible  by  ax  —  Br 

14.  Show  that,  if  a  and  B  be  any  two  ideals,  then  a  +  B  is  the 
system  of  all  numbers  of  the  form  a  -\-  ft  where  a  is  a  number  of 
a  and  j3  a  number  of  B. 

15.  Show  that,  if  a,  B  and  c  be  any  three  ideals, 

a —  [B  —  c]  ==  [a  —  B]  — c. 

16.  Show  that 

[a  +  B][a  —  6]=a6. 

17.  Show  that,  if  a  and  B  be  prime  to  each  other,  then 

a  —  B  :=  aB. 

Theorem  8.  //  a  and  B  be  any  two  ideals  prime  to  each  other, 
there  exist  a  number  a  of  a  and  a  number  |8o/B  such  that 

a  +  ^1.1 

Let        a  =  (alt a2,  •  •  •,  ar)    and   B  =  (ft,  ft,  •  •  •,  ft) . 

Since  a  and  B  are  prime  to  each  other  their  greatest  common 
divisor  is  ( 1 )  ;  that  is, 

a  +  B  =  (Op a2,  •  •  •, Or, ft, ft,  •  •  •, ft)  =  ( 1). 
But,  since  1  is  a  number  of  a  +  B,  it  must  be  a  linear  combination 

Of  CLi,a2y  ••*»«r,  ft,  ft,  "'yps', 

that  is, 

gA  +  ^20t2  H h  £rCLr  +  l?ift  +  ^2ft  H h  >7*^.  =  I, 

where  the  |'s  and  ^'s  are  integers  of  the  realm. 

But        £xax  +  |2a2  +  •  •  •  +  |rar  =  a,  is  a  number  of  a, 
and  r)xfix  +  ?72ft  +  •  •  •  +  ^sft  =  ft  is  a  number  of  B, 

and  we  have 

xThis  is  the  analogue  of  Th.  B.    See  Hilbert:  Bericht,  Satz  11. 


THE  IDEALS  OF  A  QUADRATIC  REALM.  3  I  5. 

Cor.  //  alf  ct2,  •  •  •,  am  be  ideals  whose  greatest  common  divisor 
is  (/),  then  there  exist  in  c^,  ct2,  •••,  aw  numbers  alf  a2,  •••,afft, 
respectively,  such  that 

<2i  +  a2H \-am  =  i. 

Theorem  9.  //  the  product  of  two  ideals,  a  and  b,  be  divis- 
ible by  a  prime  ideal  p,  at  least  one  of  the  ideals  is  divisible  by  p.1 

Assume  that  a  is  not  divisible  by  p.  Then  a  and  p  are  prime 
to  each  other  and  there  exists  by  Th.  8  a  number,  a,  of  a  and  a 
number,  ?r,  of  p  such  that 

a-\-7r=  1. 

Let  now  /?  be  any  number  of  b,  and  multiply  the  last  equation 
by  /?;  then 

But  aft  is  a  number  of  ab,  and  hence  by  Th.  2  of  p,  since  ab  is 
divisible  by  p.  Moreover,  irfi  is  a  number  of  p.  Hence  ft  is  a 
number  of  £ ;  that  is,  all  numbers  of  b  are  numbers  of  p,  and  b 
is  therefore  by  Th.  7  divisible  by  p. 

Cor.  1.  //  ffo  product  of  any  number  of  ideals  be  divisible 
by  a  prime  ideal,  p,  at  least  one  of  the  ideals  is  divisible  by  p. 

Cor.  2.  //  neither  of  two  ideals  be  divisible  by  a  prime  ideal, 
p,  their  product  is  not  divisible  by  p. 

Cor.  3.  //  the  product  of  two  ideals,  a  and  b,  be  divisible  by 
an  ideal,  j,  and  neither  a  nor  b  be  divisible  by  j,  then  \  is  a  com- 
posite ideal. 

If  all  the  ideals  of  a  realm  be  principal  ideals,  the  unique  fac- 
torization theorem  in  the  usual  form  holds  for  the  integers  of  the 
realm;  for,  if  a  and  (3  be  any  two  integers  prime  to  each  other 
in  the  usual  sense,  then  the  ideals  (a)  and  (/?)  are  prime  to 
each  other,  for  all  factors  of  (a)  and  (/?)  are  principal  ideals. 
Hence  the  ideal  (a,  /?)  must  be  the  unit  ideal  (1)  ;  for  (a,  /?) 
divides  both  (a)  and  (/?)  and  they  have  no  common  divisor 
other  than  (1). 

Since  (a,  £)  =  (l), 

'This  is  the  analogue  of  Th.  C.    See  Hilbert:  Bericht,  Satz  11. 


3  16  THE    IDEALS   OF   A    QUADRATIC   REALM. 

there  must  exist  two  integers,  |  and  r),  of  the  realm  such  that 

Th.  B  would  therefore  hold  for  the  integers  of  the  realm,  and 
we  have  seen  that  Th.  C,  and  hence  the  unique  factorization 
theorem,  follow  immediately.  The  converse  of  this,  that,  when- 
ever the  unique  factorization  theorem  in  its  usual  form  holds 
for  the  integers  of  a  realm,  the  ideals  of  the  realm  are  all  prin- 
cipal ideals,  is  evident;  for,  if  a,=  (alfa2,---,ar),  be  any  ideal, 
the  numbers  alt  a2,  •  •  • ,  ar  have  a  greatest  common  divisor  8,  and 
since  the  unique  factorization  law  holds  for  the  integers  of  the 
realm,  we  can  find  integers  (Chap.  V,  Th.  B,  Cor.  2)  £lt  £2,  --,£r, 
such  that 

aJi  +  a2£2  H h  a-r£r  =  8. 

Hence  we  have 

a  =  (a19 a2,  ~-,ar)  =  (a15 aa,     -,  ar, 8)  =  (8), 
a  principal  ideal.  0 

Theorem  10.  Every  ideal  can  be  represented  in  one  and  only 
one  way  as  the  product  of  prime  ideals.1 

Let  j  be  any  ideal.  If  \  be  a  prime  ideal  the  theorem  is  evident. 
If  j  be  not  a  prime  ideal,  it  has  some  divisor,  a,  different  from  j 
and  from  (1).     Then 

i=o&. 

If  a  be  not  a  prime  ideal  we  have 

a  =  axa2, 
where  ax  and  a2  are  both  different  from  a  and  ( 1 ) .     Then 

If  any  of  the  ideals  alt  a2,  6  be  not  prime,  we  factor  them,  and, 
proceeding  in  this  manner,  we  reach  finally  a  point  where  the 
factorization  can  be  carried  no  further,  for  an  ideal,  j,  is  divisible 
by  only  a  finite  number  of  ideals  (Th.  3). 

The  ideal  j  has  now  been  resolved  into  its  prime  ideal  factors. 

1  Hilbert :    Bericht,  Satz  7. 


THE    IDEALS   OF   A    QUADRATIC   REALM.  317 

Let  i=M2'-'Pr, 

where  p1} p2,---,pr  are  prime  ideals,  be  the  representation  so 
obtained.  We  shall  show  that  this  representation  is  unique. 
Suppose  that  j  could  be  represented  in  another  way  as  a  product 
of  prime  ideals,  say 

Then  pfa  • . .  pr = qxqz  . - .  q#.  3) 

Since  pt  is  a  divisor  of  the  product  qtq2  •  •  •  q8,  it  is  a  divisor 
of  one  of  its  factors  (Th.  9,  Cor.  1),  say  qlf  from  which  follows 

*>i=qi- 

Then  it  follows  from  3)  that 

p2  ...pr  =  q2  •••  qa. 

Proceeding  in  this  manner,  we  see  that  for  each  factor  in  the 
product  pjp%  •  •  •  pr  there  is  an  equal  one  in  the  product  q^2  •  •  •  q8, 
and,  reversing  the  process,  that  for  each  factor  in  the  product 
qtq2 '"  <\s,  there  is  an  equal  one  in  the  product  pxp2  •  •  •  pr,  and 
that,  if  a  factor  be  repeated  in  one  product,  it  is  repeated  exactly 
as  often  in  the  other. 

The  two  representations  are  therefore  identical,  and  the 
theorem  is  proved. 

Cor.  If  the  product  of  two  ideals,  a,  b,  be  divisible  by  an 
ideal,  m,  and  a  be  prime  to  m,  then  fc  is  divisible  by  m. 

If  we  denote  by  plt p2,'",pr  the  different  prime  ideals  that 
are  factors  of  j,  and  by  cx,e2,---,er  the  number  of  times  that 
they  are  repeated  respectively,  we  have 

It  is  convenient  sometimes  to  allow  one  or  more  of  the  expo- 
nents to  take  the  value  o,  a  =  o  indicating  that  j  does  not  contain 
pi  as  a  factor.  It  is  evident  that  an  ideal  j  is  then  and  only  then 
divisible  by  an  ideal  b  if  every  prime  ideal  which  divides  b  occurs 
to  at  least  as  high  a  power  as  a  factor  in  j  as  it  does  in  b. 

Every  divisor  of  j  has  therefore  the  form 

b  =  p1m>V2m*'-'prmr,  4) 


3  l8  THE  IDEALS  OF  A  QUADRATIC  REALM. 

where  nti^ei;   i=i,2,--,rJ 

and  every  ideal  of  the  form  4)  is  a  divisor  of  j.  If  we  let  m* 
run  through  the  *<  +  x  values,  o,  I,  ••-,  eiy  and  do  this  for  each  of 
the  exponents  mls  m2,  •••,  mr,  we  obtain 

different  sets  of  values  for  these  exponents,  and  each  of  these 
sets  gives  a  different  divisor  of  j.  The  number  of  divisors  of  j 
is  therefore   (e1  +  1)  (e2  -f-  1)  •••  (er  -\-  1). 

If  i  =  p1mi^2m2  •••^rmr, 

and  f)  =  p^/2  •  •  •  prnr, 

where  pls  p2,  •••,  pr  are  different  prime  ideals,  be  any  two  ideals, 
the  ideal 

where  gi  is  the  lesser  of  the  two  exponents  mi  and  m(i  =  l,  2, 
•••,r),  is  the  greatest  common  divisor  of  j  and  f). 
The  ideal 

where  U  is  the  greater  of  the  two  exponents  mi  and  «<(t=l,  2, 
•••,r)  is  the  least  common  multiple  of  j  and  f). 

We  see  from  this  representation  of  the  greatest  common  divisor, 
g,  of  j  and  §  that,  of  all  common  divisors  of  }  and  %  g  has  the 
greatest  norm,  and  that  the  quotients,  j/g  and  Ij/g,  are  prime  to 
each  other  (see  p.  18). 

Theorem  ii.  If  a.  and  m  be  any  two  ideals,  there  exists  a 
number,  a,  of  a  such  that  the  quotient  (a) /a  is  prime  to  m. 

For  example,  if  a,=  (2,  1  -f-V — 5),  and  m,=  (3,  1  +V — 5), 
be  the  given  ideals,  then  a  =  2  satisfies  the  requirements  of  the 
theorem,  for 


THE    IDEALS   OF   A   QUADRATIC   REALM.  319 

that  is  easily  seen  to  be  prime  to  (3,  1  -)-y — 5). 

If  a,  =  (2,  i+V^5),  and  m,=  (1  +V=I5),  be  the  given 
ideals,  then  a,  =  2  +  1 +Vir5,  =  3 +V=r5,  satisfies  the  re- 
quirements of  the  theorem  for 

(3  +  v^      ,    3  +  v~s) 
(2,i+v=T)    (7'3  +  v    sh 

that  is  prime  to  (1  +V — 5). 

For  the  actual  determination  of  a  in  general  see  Chap.  XII,  §  7. 

We  proceed  now  to  prove  the  theorem. 

The  truth  of  this  theorem  for  the  case  where  m  is  any  prime 
ideal  p  is  at  once  evident.  For,  if  there  did  not  exist  a  number, 
a  of  a  such  that  (a)/a  is  not  divisible  by  p,  then  all  numbers  of 
a  would  belong  to  ap  and  by  Th.  7  a  would  be  divisible  by  ap, 
which  is  impossible.  To  prove  the  theorem  for  the  general  case, 
let  the  different  prime  factors  of  m  be  plt  p2,  •••,pmi  and  form 
the  products 

<*!  =  ap2  •  •  •  pm,  a2  =  ap!p3  •  •  •  pm,  •  •  • ,  am  =  apx  •  •  •  pm-x, 
which  consist  of  a  multiplied  in  turn  by  the  combinations  of 
Pi>p2>  ••-,Pm  taken  m — 1  at  a  time.  Let  alt  a2,  -',am  be  num- 
bers of  alt  a2,  •••,  aOT  respectively,  such  that  (aO/c^,  («2)/a2, 
•  •  •,  (am)/am  are  prime  respectively  to  plt  p2,  •  •  •,  pm,  the  existence 
of  such  numbers  having  been  proved  above  since  plf  p2,  •••,  pm 
are  prime  ideals.     Then 

a  =  a1-\-a2-{ \-am, 

is  the  required  number;  for  a  is  divisible  by  a,  since  ax,a2,  -",am 
are  all  divisible  by  a,  cu  being  divisible  by  a,  whence  all  numbers 
of  cu  belong  to  a;  moreover,  a  is  not  divisible  by  any  of  the  m 
products 

api,  ap2,  '-,apm, 

as,  for  example,  aplt  since  a2,as,  •••,am  are  all  divisible  by  aplf 
but  ax  is  not  divisible  by  apx. 

It  is  evident,  therefore,  that  the  quotient  (a) /a  is  divisible  by 


320  THE  IDEALS  OF  A  QUADRATIC  REALM. 

none  of  the  prime  factors  plfp2,  •••ipm  of  m,  and  hence  is  prime 
to  m. 

Hence  a  is  the  required  number. 

Theorem  12.  In  every  ideal,  a,  there  exist  two  numbers,  alf 
a2,  such  that 

a=(a1,a2); 

that  is,  such  that  a  is  the  greatest  common  divisor  of  (a±)  and 
(a2). 

Let  at  be  any  number  of  a. 

By  Th.  11  there  exists  in  a  a  number,  a2,  such  that  the  quotient 
{a2)/a  is  prime  to  (ax)  ;  or,  in  other  words,  such  that  the  greatest 
common  divisor  of  (ax)  and  (a2)  is  a. 

But,  since  a  is  the  greatest  common  divisor  of  (at)  and  (a2), 
it  contains  all  and  only  numbers  of  the  form 

P&  +  p2a2, 

where  f}v  fi2  are  any  integers  of  the  realm.     Hence 

a=(alt  a2). 

The  truth  of  this  theorem  is  at  once  evident  for  quadratic  realms  for 
we  have  shown  (Th.  1)  the  existence  in  every  ideal,  a,  of  a  quadratic 
realm  of  two  numbers  ha  h  such  that  a=  («i,  t2).  The  proof  in  the  above 
form  has  been  given,  however,  as  it  applies  to  the  general  realm  of  the 
Mth  degree;  see  Hilbert :  Bericht,  Satz  12. 

The  following  theorem  is  given  not  only  for  its  own  interest 
but  because  from  it  we  obtain  a  new  proof  of  Th.  11  that  is  not 
dependent  upon  the  unique  factorization  theorem.  Dedekind 
makes  the  unique  factorization  theorem  depend  upon  Th.  13 
(see  Dirichlet-Dedekind,  §  178,  IX). 

Theorem  13.  //  the  ideal  a  be  divisible  by  none  of  the  ideals 
Ci»  c2,  •••,(:„,  then  there  is  a  number,  a,  of  a  that  is  contained  in 
none  of  the  ideals  clf  c2,  ••-•,  c«. 

If  a  should  be  a  principal  ideal,  the  theorem  is  evident.  Also, 
if  there  should  be  only  a  single  ideal,  c,  the  theorem  holds,  for,  if 
all  numbers  of  a  were  divisible  by  c,  a  would  be  divisible  by  c, 


THE    IDEALS   OF   A   QUADRATIC   REALM.  32 1 

which  is  contrary  to  the  original  hypothesis.  We  shall  now  prove 
that,  if  the  theorem  hold  for  n  <  r  it  holds  for  n  =  r,  and  hence, 
since  it  is  true  for  n=i,  it  holds  in  general.  To  each  of  the 
ideals  clf  c2,  •••,  cr,  as  c«,  there  corresponds  an  ideal  h8  such  that 

ab8  =  a  —  c„ 

where  h8  is  evidently  different  from  (i). 

The  ideal  a  is  divisible  by  none  of  the  r  products 

dblt  ah2,  •  •  •,  abr,  5) 

since  all  of  the  b's  are  different  from  (1). 

But  each  one  of  the  c's  divides  one  of  these  products.  Hence, 
if  we  can  prove  the  existence  of  a  number  of  a,  which  belongs  to 
none  of  the  products  5),  this  number  will  be  the  desired  number  a, 
for  if  a  were  divisible  by  c«,  it,  being  divisible  by  a,  would  be 
divisible  by  the  least  common  multiple  of  a  and  c«;  that  is,  ab8. 
We  have  now  two  cases  to  consider  according  as  the  ideals  hlf  62, 
•••,  fcr  are,  or  are  not,  prime  each  to  each.  If  they  be  not  prime 
each  to  each,  some  pair  of  them,  say  hlf  B2,  must  have  a  greatest 
common  divisor,  ht  +  ft2,  that  is  different  from  ( 1 ) . 

Then  a  is  not  divisible  by  a(Bx  +  b2),  and  hence,  according  to 
our  assumption  that  the  theorem  holds  for  n  <  r,  there  exists  in 
a  a  number,  a,  that  is  divisible  by  none  of  the  r —  1  ideals 

a(B1  +  B2),ab3,  --,cibr, 

and  hence  also  is  not  divisible  by  ab1  and  ab2,  since  they  are  divis- 
ible by  a(Bx-|-B2).  Therefore  a  is  not  divisible  by  any  of  the 
c's.  We  must  consider  now  the  case  where  the  r  ideals,  blt  b2,  •  •  •, 
Br  are  prime  each  to  each. 

Each  of  these  ideals,  as  6„  is  prime  to  the  product,  $«,  of  all 
the  remaining  ones,  and,  since  they  are  all  different  from  (1), 
Ijs  is  not  divisible  by  &,.  Hence  at}8  is  not  divisible  by  ab8,  and 
there  is  therefore  a  number  a8,  in  a$8  that  is  not  divisible  by  ab8. 

The  number  a,  =  at  +  a2  -\ \-ar,  where  alf  a2,  •  •  •,  ar  are 

numbers  of  a^lf  at)2,  •  •  •,  df)r  respectively,  is  a  number  of  a,  for  each 
21 


322  THE    IDEALS    OF    A    QUADRATIC    REALM. 

of  the  numbers  alfa2>  -",ar  is  a  number  of  an  ideal  divisible  by 
a,  and  is  therefore  a  number  of  a. 

Moreover,  a  is  divisible  by  none  of  the  r  products  ablf  ab2,  •  •  • , 
ahr;  for,  since  the  ideals  §2,  §8,  •  •  •,  §r  are  all  divisible  by  i)lf  all  the 
ideals  ai)2,  •••,af)r  are  divisible  by  ab1}  and  hence  a2, a3,  ---jCtr  are 
numbers  of  qBj. 

But  ax  is  not  a  number  of  abx,  and  hence  a  is  not  a  number 
of  abj. 

In  like  manner  it  may  be  proved  that  a  is  divisible  by  none  of 
the  ideals  ab2,  abs,  •  •  •,  abr. 

Hence  a  is  the  number  sought. 

Second  Proof  of  Theorem  II.1 

If  m=(i),  every  number  of  a  satisfies  the  requirement  of 
Th.  ii. 

If  m+C1)*  let  c,,  c2,  •••,c,l  be  all  the  ideals  different  from  a 
that  divide  am  and  are  divisible  by  a. 

By  Th.  3  these  ideals  are  finite  in  number  and  hence  there  is 
in  a  a  number,  a,  that  is  divisible  by  none  of  them  (Th.  13). 

Hence  the  greatest  common  divisor,  am  +  (a),  of  am  and  (a) 
is  different  from  all  the  c's.  But  am -f  (a)  divides  am  and  is 
divisible  by  a,  and  the  only  ideal  different  from  the  c's,  that  has 
this  property,  is  a. 

Hence  am-\-(a)=a,  6) 

or,  what  is  the  same  thing,  (a) /a  is  prime  to  m. 
From  6)  it  follows  at  once  that 

am —  (a)  =m(a). 

1  Dirichlet-Dedekind  :  §178,  X. 


CHAPTER   XII. 

Congruences  whose  Moduli  are  Ideals.1 

§  i.     Definition.     Elementary  Theorems. 

If  the  difference  of  two  integers,  a  and  /?,  be  a  number  of  the 
ideal  o,  we  have  said  that  a  was  congruent  to  /?  with  respect  to 
the  modulus  a,  and  have  denoted  this  fact  by  writing 

a  =  /?,  mod  a.  i) 

In  particular,  if  a  be  a  number  of  a,  we  write 

a  =  o,  mod  a. 

The  appropriateness  of  these  symbolic  expressions  is  made 
evident  by  Chap.  XI,  Th.  7 ;  for  from  it  we  see  that  the  necessary 
and  sufficient  condition  for  a  —  fi  to  be  a  number  of  a  is  that  it 
shall  be  divisible  by  a.  These  expressions  are  capable  of  many 
of  the  transformations  to  which  ordinary  congruences  between 
rational  integers  can  be  subjected.     The  congruence  1)  leads  to 

a  —  /?  =  o,  mod  a,  2) 

and  conversely  2)  leads  to  1). 

The  following  deductions  will  be  seen  to  correspond  number 
for  number  to  those  given  in  the  case  of  rational  integers  (Chap. 
Ill,  §1).  Their  proofs  are  so  simple  that  they  will  be  left  to 
the  reader.     For  them  we  fall  back  upon  our  original  definition  of 

a  =  /?,  mod  a, 

as  meaning  that  a  —  (3  is  a  member  of  a,  or,  what  is  the  same 
thing,  that  the  principal  ideal  (a  —  /3)  is  divisible  by  a.  Observe 
the  similarity  between  this  and  the  method  employed  in  the  case 
of  rational  integers,  where  we  made  use  of  our  original  defi- 
nition of 

a  =  fr,  mod  m, 

1  Hilbert :  Bericht,  Cap.  III. 

323 


324  CONGRUENCES    WHOSE    MODULI    ARE    IDEALS. 

as  meaning  that  a  —  b  is  divisible  by  m. 

i.  //  ol  =  /3,  mod  a, 

and  /?  =  y>  m°d  a, 

then  a  =  y,  mod  a; 

for,  if  a  —  (3  and  /?  —  y  be  numbers  of  a,  a  —  (S  -\-  fi  —  y,  =  a  —  y, 
is  a  number  of  a. 

The  infinite  system  of  integers  of  the  realm  which  are  con- 
gruent to  a  given  integer,  and  hence  each  to  each,  mod  a,  are  said 
to  form  a  number  class,  mod  a. 


ii.  // 

as=/3,  mod  a, 

and 

yE=8,  mod  a, 

then 

a  ±  y  S3  j3  ±  8,  mod  a. 

iii.  // 

ass/},  mod  a, 

then 

^a  =  /x/?,  mod  a. 

iv.  // 

a  =  /?,  mod  a, 

and 

y  =  8,  mod  a, 

then 

ay  =  /?8,  mod  a; 

and,  in  particular, 

if 

a  =  /?,  mod  a, 

fftlff  a*  =  /3«,  mod  a. 

v.  //  /(*)  =  a0*»  +  a^-1  +  •  •  •  +  an, 

&£  a  polynomial  in  x,  whose  coefficients  are  any  integers  of  the 
realm,  and  if 

/3==y,  mod  a, 

then  /(/?)  =/(y),  mod  0. 

vi.  //  fia  =  (xp,  mod  a,  3) 

Men  a  =  /?,  mod  a/b, 

where  b  W  f/i<?  greatest  common  divisor  of  (/*)  and  a. 


CONGRUENCES    WHOSE    MODULI    ARE    IDEALS.  325 

For  let  (fi)  =  bm  and  a  =  bb,  where  m  and  b  are  prime  to  each 
other;  then,  since  fi[a  —  (3]  is  a  number  of  a,  bm(a  —  /3)  is  divis- 
ible by  bb. 

Hence  m(a  —  £)  is  divisible  by  b,  and  therefore,  since  m  is 
prime  to  b,  (a  —  /?)  is  divisible  by  b  (Chap.  XI,  Th.  10,  Cor.). 
We  have,  therefore,  since  b  =  a/b. 

a  =  /?,  mod  a/b. 

In  particular,  if  ji  be  prime  to  a,  then 

aE=/?,  mod  a. 

Hence  in  this  case  the  congruence  i)  may  be  divided  by  /x. 

This  indeed  is  an  immediate  consequence  of  the  fact  that  the 
greatest  common  divisor  of  (jx)  and  a  is  (i)  ;  for  then  there  is 
a  number  /*£  of  (./a)  and  a  number  y  of  a  such  that 

/^  +  y=i; 

that  is,  there  exists  an  integer  £  such  that 

/*!e=i,  mod  a.  4) 

Multiplying  the  congruence  3)  by  £,  we  obtain 

a  =  /J,  mod  a. 

Conversely,  if  there  exists  a  number  |,  which  satisfies  the  con- 
gruence 4),  the  greatest  common  divisor  of  (fx)  and  a  is  (1)  ; 
that  is,  (/x)  is  prime  to  a. 

vii.  If  ct^/3,  mod  a 

awe?  b  be  a  divisor  of  a,  then 

a  =  j3,  mod  b. 

viii.  If  a  =  /3  with  respect  to  each  of  the  moduli  au  a2,  •••,(!«, 
*/*£»  az==P,  mod  I, 

zvhere  I  w  ffc  /<?a,rt  common  multiple  of  alt  o2,  •  ••,(!«. 

ix.  //  a  =  /?,  mod  a, 

f/z^n  (a)  and  (/?)  &az^  £/^  saw*?  greatest  common  divisor  with  a; 


326  CONGRUENCES   WHOSE    MODULI   ARE   IDEALS. 

that  is,  all  numbers  of  the  same  number  class,  mod  a,  have  the 
same  greatest  common  divisor  with  a. 

Let  b  be  the  greatest  common  divisor  of  (a)  and  a.  Then, 
since  b  is  a  divisor  of  a,  we  have  by  vii 

a  ===/?,  mod  b. 

But  a  =  o,  mod  b, 

and  hence  P  =  o,  mod  b. 

In  particular,  if  any  number  of  a  class,  mod  a,  be  prime  to  a, 
then  all  numbers  of  this  class  are  prime  to  a. 

§  2.  The  Norm  of  an  Ideal.  Classification  of  the  Numbers 
of  an  Ideal  with  respect  to  Another  Ideal. 

If  we  separate  the  integers  of  a  realm  into  classes  with  respect 
to  an  ideal,  a,  of  the  realm,  putting  two  integers  into  the  same  or 
different  classes  according  as  they  are  congruent  or  incongruent 
to  each  other  with  respect  to  a,  then  the  number  of  these  classes  is 
called  the  norm  of  a,  and  is  denoted  by  n[a\. 

This  definition  of  the  norm  of  an  ideal  is  seen  to  be  in  accord- 
ance with  the  principal  property  possessed  by  the  absolute  value 
of  the  norm  of  an  integer.  We  shall  show  later  that  the  original 
definition  of  the  norm  of  an  integer  as  the  product  of  an  integer 
by  its  conjugate  has  also  its  analogue  in  the  case  of  ideals. 

A  system  of  numbers  formed  by  selecting  one  from  each  of  the 
classes  formed  as  above  with  respect  to  an  ideal,  a,  is  called  a 
complete  system  of  incongruent  numbers,  mod  a,  or  a  complete 
residue  system,  mod  a.  There  are  evidently  in  such  a  system 
exactly  n[a]  numbers. 

Instead  of  separating  all  the  integers  of  a  realm  into  classes 
with  regard  to  their  congruence  with  respect  to  an  ideal,  we  may 
consider  simply  the  numbers  of  a  single  ideal,  a,  and  put  two  of 
these  numbers,  alf  a2,  into  the  same  or  different  classes  with 
respect  to  an  ideal,  b,  according  as  we  have 

OLi^a.2,  mod  b,  or  ax^p.a2,  mod  b. 

We  shall  denote  by  the  symbol   {a,  b}   the  number  of  such 
classes  into  which  the  numbers  of  a  fall  with  respect  to  b.1 
1  See  Dirichlet-Dedekind  :  §  171. 


CONGRUENCES    WHOSE    MODULI    ARE   IDEALS.  327 

Evidently  {a,  b}  is  not  greater  than  w[b],  since  a  does  not  com- 
prise all  integers  of  the  realm  k  unless  a=(i),  in  which  case 
{(i),6}  =  «[6]. 

It  will  be  interesting  to  make  use  of  this  classification  of  the 
numbers  of  one  ideal  with  respect  to  another  ideal  to  prove  an 
important  theorem  (see  p.  336)  and  we  proceed  now  to  prove  the 
following  relations : 

i.  {a,  *}  =  {a,  a  —  b}. 

ii.  {a,  h}  =  {a  +  h,  b}. 

iii.  {a(v),  b(^)}  =  {a,  b}. 

iv.  {a,  c}  =  {a,  &}{&,  c}, 

where  a  is  a  divisor  of  b,  and  b  a  divisor  of  c. 

i.  To  prove  {a,  &}  =  {a,  a  —  &}. 

We  observe  that  a  —  b,  the  least  common  multiple  of  a  and  b, 
is  composed  of  all  numbers  common  to  both  a  and  b. 
Hence,  if  alt  a2  be  two  numbers  of  a  such  that 

0^  =  32*  m°d  b, 

that  is,  such  that  ax  —  a2  is  a  number  of  b,  then,  since  at  —  a2  is 
also  a  number  of  a,  it  must  be  a  number  of  a  —  b,  and  therefore 

ax=a2,  mod  a  —  b. 
Conversely,  if 

ax  =  a2,  mod  a  —  b, 

then  ax  —  a2  is  a  number  of  b;  that  is, 

ax^a2,  mod  b. 

Hence  any  two  numbers  of  a,  that  are  congruent  to  each  other 
with  respect  to  b,  are  congruent  to  each  other  with  respect  to 
a  —  b  and  vice  versa.     Therefore  we  have 

{a,  h}  =  {a,  a-h}. 

ii.  To  prove  {a,  b}  =  {a-f  b,  b}. 

Let  a!,a2,  ••-,am  (m  =  {a,  b})  1) 


328  CONGRUENCES    WHOSE    MODULI    ARE    IDEALS. 

be  a  complete  system  of  incongruent  numbers  of  a  with  respect  to 
b.  Then  every  number  of  a  +  8  is  congruent  to  one  of  these 
numbers  with  respect  to  ft,  for  all  numbers  of  a  +  B  can  be 
written  in  the  form  a  +  /3,  where  a  is  a  number  of  a  and  /3  a 
number  of  B.     And  from 

a^di,  mod  fc, 

where  a*  is  one  of  the  numbers  1),  we  have 

a-\-  /3  =  oii,  mod  B, 

since  /?  =  o,  mod  ft. 

Moreover,  since  a  +  ft  contains  all  the  numbers  of  a,  some 
numbers  of  a  +  &  will  be  congruent  to  each  one  of  the  integers 
of  the  system  1),  mod  ft.     Hence 

{a,  6}  =  {a +  6,  6}. 
iii.  To  prove 

{aW,BW}  =  {aJ}. 

Let  a1,a2,--,am(m={a,  &}) 

be  a  complete  system  of  incongruent  numbers  of  a  with  respect  to 

b,  then  G^iy,  a2r),  •  •  • ,  am?/ 

form  a  complete  system  of  incongruent  numbers  of  a(>y)  with 
respect  to  the  mod  6(77)  ;  for  they  are  all  incongruent,  mod  &(*/), 
to  each  other,  since,  if 

agr]==ahr),  mod  b(^), 

then  agz==ah,  mod  &, 

which  is  impossible.  Furthermore,  every  number  of  a(rj)  is  con- 
gruent to  one  of  these  integers,  mod  b(^),  for,  if  ay  be  any  num- 
ber of  a(rj),  and 

a  =  at,  mod  b, 

then  (a —  a.i)r)  is  a  number  of  &(??),  and  hence 

arj^ai-rj,  mod  16(17). 

Hence  {<*(,),  &«}  =  {a,  b}. 


CONGRUENCES    WHOSE    MODULI   ARE   IDEALS.  329 

iv.  To  prove  that,  if  a  be  a  divisor  of  b  and  b  a  divisor  of  c,  then 

{a,  c}  =  {a,  h}{h,  c}. 

Let  a1,a2,'~tam(tn={a,  fy)  2) 

be  a  complete  system  of  incongruent  numbers  of  a  with  respect 
to  the  modulus  b,  and  let 

ft, ft,  —  ,ft(n  =  {6,  c})  3) 

be  a  complete  system  of  incongruent  numbers  of  b,  mod  c.     We 
shall  show  that  the  mn  numbers 

[  r=  1,2,  ••-,*» 

Zr  +  Psi  4) 

L$  =  1,2,  •••,M 

which  are  all  evidently  numbers  of  a,  form  a  complete  system  of 
incongruent  numbers  of  a,  mod  c. 

The  numbers  4)  are  incongruent  each  to  each,  mod  c;  for,  if 

aa  +  Pb  =  ac  +  pd,  mod  c,  5) 

then,  since  b  is  a  divisor  of  c, 

aa  -f  fii  sb  ac  +  /?<?,  mod  b, 

and  hence,  since  fib  and  pa  are  numbers  of  b, 

aa==ac,  mod  b, 

which  is  impossible  unless  aa  =  ac.     But,  if  aa  =  ac,  then  from 
5)  we  have 

pb==pd,  mod  c, 

which   is   impossible.     Hence   the   numbers  4)    are   incongruent 
each  to  each,  mod  c. 

Moreover,  every  number,  a,  of  a  is  congruent  to  some  one  of 
the  numbers  4),  mod  c;  for  suppose 

a^a-x,  mod  b, 

where  at  is  one  of  the  numbers  2),  then  a  —  a»  is  a  number  of  b, 
and  we  have 

a  —  a<ss/b,  mod  c, 


330 


CONGRUENCES   WHOSE    MODULI   ARE   IDEALS. 


where  ph  is  one  of  the  numbers  3),  and  hence 

a  =  a*  +  fa,  mod  c, 

where  a«  +  fa  is  one  of  the  numbers  4). 

The  numbers  of  a  complete  system  of  incongruent  numbers  of 
a,  mod  c,  are  therefore  exactly  mn  in  number,  and  hence 

{a,  c}  =  {a,  &}{*>,  c}. 

Theorem  i.  If  ilt  =  a1o)1  -\-  a2w2,  i2,  =  &!&>!  +  b2<a2,  £?£  a  &a«'j 
0/  f&£  wfea/  a,  £/z£  absolute  value  of  the  determinant  of  the  coeffi- 
cients alf  a2,  blf  b2  is  equal  to  the  norm  of  a;  that  is, 


«[a]  = 


Let 


"1     "2 


where  a<olt  b(o±  +  cw2  is  a  canonical  basis,  a  and  c  being  taken 
positive.     Since 


1      ^2 
it  is  sufficient  to  show  that 


a     o 

b     c 


=  ac      (Chap.  XI,  §  2), 


w[q]  =ac. 
In  the  expression 

Umx  -j-  ^a>2  6) 

let  w  run  through  the  values  o,  1,  •••,  a — 1,  and  v  through  the 
values  o,  1,  •••,  c — 1.  We  shall  show  that  the  ac  numbers  so 
formed  constitute  a  complete  system  of  incongruent  numbers  with 
respect  to  a.  m  They  are  incongruent  each  to  each  with  respect 
to  a;  for,  if  u1<o1  +  vxoi2  and  u2&x  +  v2o>2  be  any  two  of  them,  and 

Wjtoi  -\-  v1(o2  ss  WjCDj  -f-  z>2w2,  mod  a, 

then  {ux  —  u^^-^- (v1  —  v2)(o2^o,  mod  a, 

and  hence,  since  c  is  the  greatest  common  divisor  of  the  coeffi- 
cient of  <o2  in  all  numbers  of  a, 


1Hilbert:  Bericht,  Satz  19. 


CONGRUENCES    WHOSE    MODULI    ARE    IDEALS.  33 1 

vx  —  v2  =  0,  mod  c. 

But  vx  and  v2  are  both  less  than  c,  hence 

vx  =  v2. 
It  follows  that 

(ux —  u2)a>x===o,  mod  a, 

and  hence,  since  a  is  the  greatest  common  divisor  of  the  coeffi- 
cients of  o)x  in  all  numbers  of  a  in  which  the  coefficient  of  <o2  is  o, 

ux —  u2==o,  mod  a. 

But  ux  and  u2  are  both  less  than  a,  hence 

ux  =  u2. 

ThUS  Ux0)x  -f"  ^iW2  =  W^  +  Z>2W2> 

and  the  numbers  6)  are  incongruent  each  to  each  with  respect  to 
a.  Moreover,  every  integer  of  the  realm  is  congruent  to  one  of 
the  numbers  6)  with  respect  to  a.     For,  let 

(0  =  txoix  -j-  t2oi2 

be  any  integer  of  the  realm,  and  let 

t2  =  mc  +  r2, 

where  m  and  r2  are  rational  integers  and  r2  satisfies  the  conditions 

ogr2<c. 

Also  let  tx  —  mb  =  na-\-rx, 

where  n  and  rx  are  rational  integers  and  rx  satisfies  the  conditions 

0  g  rx  <  a. 
Then 

*1W1  +  ^2W2  =   {™b  +  na  +  ri)Wl  +    (  WC  +  r2)W2 

=  waw!  -j-  m(bo)x  +  ^w2)  +  riwi  +  r2w2> 

and  hence  f  jo^  -J-  t2w2  h  rxo)x  +  r2w2,  mod  a. 

But  r^  +  r2w2  is  one  of  the  numbers  6). 

Hence  every  integer  of  the  realm  is  congruent  to  one  of  these 


332 


CONGRUENCES   WHOSE    MODULI   ARE   IDEALS. 


numbers  with  respect  to  a,  and  therefore,  since  they  are  ac  in 
number 

n[a]  =  ac. 


Hence 


n[a]  = 


K     bi\ 


From  this  theorem  we  see  that  the  norm  of  an  ideal  is  always 
finite. 

Ex.     Since  7,  3  +  y/  —  5  is  a  basis  of  the  ideal  (7,  3  +  V  —  5)> 

7    o 


»(7i  3  +  V—  5! 


3     « 


=  7. 


In  the  case  of  non-principal  ideals,  we  shall  omit  [  ]  and  write  merely 
n  before  the  symbol  to  denote  the  norm,  as  in  the  example  just  given. 

Cor.  1.     Since,  if  a1w1  +  o2o>2,  b1a)1  +  b2<o2  be  a  basis  of  a,  then 

aiwi'  +  a2«>2>  friwi'  +  ^2^2'  W  a  basis  of  a'  (Chap.  XI,  §  j),  we  have 


«[v]  = 


a,    an 


h  K 


n  [a] . 


Cor.  2.     If  (a)  be  a  principal  ideal,  where  a  is  a  rational  in- 
teger, then 

n[(a)]=a2; 

for  awj,  aa>2  is  a  basis  of  (a),  and  hence 

a    o 


»[(*)]  = 


O    a 


=  <r. 


We  can  prove  by  this  method  that  the  norm  of  any  principal 
ideal  (a)  is  equal  to  the  absolute  value  of  the  norm  of  the 
integer  a  which  defines  (a)  ;  that  is 

n[(a)]  =  \n[a]\. 

But  a  simpler  proof  can  be  found,  based  upon  a  theorem  to  be 
given  later. 

Cor.  3.    //  a,=  (ax<ax  +  a2w2>  &iwi  +  b2<o2),  be  any  ideal  and 

a\     a2 

K    K  =4a]' 

then  a1<a1  -j-  a2w2,  bxtar  -J-  fr2w2  £y  a  basis  of  a. 


CONGRUENCES    WHOSE    MODULI    ARE   IDEALS.  333 

Theorem  2.  If  a  =  bc,  zvhere  b  and  c  are  any  ideals,  there  are 
exactly  n[c]  numbers  of  a  complete  system  of  incongruent  num- 
bers, mod  a,  which  are  divisible  by  b. 

Let  rify»""»Y»['i  7) 

be  a  complete  system  of  incongruent  numbers,  mod  c,  and  let  /? 
be  a  number  of  b  such  that  (/?)/&  is  prime  to  c  (Chap.  XI,  Th. 
11).     The  numbers 

are  incongruent  each  to  each,  mod  a;  for,  if 

pyh==f$yi,  mod  a, 

then  yft  =  y<,mod  c  (§1,  vi), 

which  is  impossible. 

Moreover,  every  integer,  @lt  divisible  by  b  is  congruent,  mod 
a,  to  some  integer  of  the  form  n/3,  for  since  b  is  the  greatest 
common  divisor  of  a,=  (alfa2),  and  (/?),  we  have 

K=(a1,a2,p), 

whence,  since  ft  is  a  number  of  b,  it  follows  that 

where  |x,  |2  and  fx  are  integers  of  the  realm,  and  hence 

P1==fi(3,  mod  Q. 

But  every  integer  of  the  form  fif3  is  congruent,  mod  a,  to  some 
one  of  the  numbers  8)  ;  for  fi  is  congruent  to  some  one,  say,  -/», 
of  the  numbers  7),  mod  c,  and  from 

fj.^yif  mod  c, 
it  follows  easily  that 

fifx==l3yi,  mod  a. 

Since,  also,  every  integer  congruent  to  one  of  the  numbers  8), 
mod  a,  is  divisible  by  b  (§1,  vii),  and  the  numbers  8)  are  «[c]  in 
number,  there  are  in  every  complete  system  of  incongruent  num- 
bers, mod  a,  exactly  n[c],  =  n[a]/n[b],  numbers  that  are  divis- 
ible by  b. 


334  CONGRUENCES    WHOSE    MODULI    ARE    IDEALS. 

Theorem  3.  The  norm  of  the  product  of  two  ideals,  a,  b,  is 
equal  to  the  product  of  their  norms.1 

Let  a  be  a  number  of  a  such  that  the  quotient  (a) /a  is  prime 
to  6  (Chap.  XI,  Th.  11). 

Let  CLi,CL2>  •">an[a]  9) 

and  Pi,P2,-",Pnif>i  10) 

be  complete  systems  of  incongruent  numbers  with  respect  to  a 
and  b,  respectively.     Then  the  w[ct]n[b]  numbers  of  the  form 

where  £  and  17  run  through  the  values  9)  and  10),  respectively, 
form  a  complete  system  of  incongruent  numbers  with  respect  to 
ah,  and  hence  are  n[ab]  in  number. 

To  show  this  it  is  necessary  and  sufficient  to  show  first  that 
no  two  of  the  integers  11)  are  congruent  to  each  other  with 
respect  to  the  modulus  ab,  and  second  that  every  integer  of  the 
realm  is  congruent  to  one  of  them  with  respect  to  ab. 

Let  api  -\-  ai  and  a/3j  -f-  am  be  any  two  of  the  integers  11). 

If  afii  +  ai  =  a/3j  +  am,  mod  ab,  12) 

then  a(pi  —  pj)-\-a% — am  =  o,  mod  a, 

and  hence,  since 

a  ((3i  —  fij )  =  o,  mod  a, 

we  have  ai  —  am  =  o,  mod  a, 

whence  a.i  =  am. 

Then  from  12)  it  would  follow  that 

a  (pi  —  pi)  =  o,  mod  ab, 

and  hence,  since  (a)  +  ab  is  a, 

/3i  —  pj  =  o,  mod  b, 

which  is  impossible  unless 

pi  =  pj. 
1Hilbert:  Bericht,  Satz  18. 


CONGRUENCES    WHOSE    MODULI    ARE   IDEALS.  335 

Therefore  12)  is  impossible  and  the  integers  11)  are  incon- 
gruent  each  to  each,  mod  ah.  Moreover,  if  to  be  any  integer  of 
the  realm,  we  have 

a«  =  w,  mod  a,  13) 

where  a8  is  one  of  the  integers  9). 

Now  from  13)  it  follows  that  w  —  a8  is  divisible  by  a.  But 
every  integer  of  a  complete  residue  system,  mod  ah,  that  is  divis- 
ible by  a  is  congruent  to  one  of  the  integers 

apltap2i  •••,a0„[6],  14) 

mod  ah  (Th.  2)  ;  that  is,  the  integers  14)  are  representatives  of 
all  and  only  those  incongruent  number  classes,  mod  ah,  whose 
numbers  are  divisible  by  a. 
Hence  we  have 

w  —  a8^a(3r,  mod  ah, 

whence  <o  =  a/?r -f-ag,  mod  ah, 

where  af3r  +  cl8  is  one  of  the  numbers  11). 

The  numbers  11)  form  therefore  a  complete  system  of  incon- 
gruent numbers,  mod  ah,  and  hence  i 

n[ah]  =n[a]n[h]. 

A  complete  system  of  incongruent  numbers,  mod  ah,  fall  int 
11  [a]  classes  each  containing  n[h]  numbers,  such  that  the  numbers 
of  each  class  are  congruent  each  to  each,  mod  a,  but  the  numbers 
of  any  class  are  incongruent  to  all  those  of  any  other  class,  mod  a. 
We  may  arrange  these  classes  as  follows: 

aft  +  att  ap2  +  alf  •  •  •,  a0„[6]  +  alt 


ft 


aPi  +  3n[a],a/?2  +a«[a]»  ••  •><*/?«[*]  -f-a»[a], 
where  a,alfa2,  •  •  -,a» [«],&,&>  "m9pm*i  are  as  defined  above. 

It  will  be  seen  that  the  numbers  of  each  row  are  all  and  only 
those  of  the  complete  system  of  incongruent  numbers,  mod  ah, 
that  are  congruent  to  each  other,  mod  a. 


33^  CONGRUENCES    WHOSE    MODULI    ARE   IDEALS. 

There  are,  therefore,  exactly  n[b]  numbers  of  a  complete 
residue  system,  mod  ab,  that  are  congruent  to  any  given  number, 
mod  a.  In  particular  there  are,  as  we  have  already  seen,  exactly 
n[b]  numbers  of  a  complete  residue  system,  mod  ab,  which  are 
divisible  by  a. 

It  will  be  interesting  to  obtain  by  means  of  the  development  of   §  2 
another  proof  of  the  above  important  theorem. 
We  begin  by  proving  that 

{a,  ab}  =*[&].      ' 

Let  a  be  a  number  of  0  such  that  ah  -j-  (a)  =  a;  then* 

ab—(a)=b(a), 

for  the  least  common  multiple  of  two  ideals  is  equal  to  their  product 
divided  by  their  greatest  common  divisor.    We  have  now 

{(a),  ab}  =  {(a)-f  ab,  ab}    (§2,  ii) 
=  {a,  ab}, 

and  also  {(a),  ab}  =  {(a),  (a)  —  ab}     (§2,  i) 

=  {(«),  (a)b} 

=  {(1),  b}  (§2,  iii) 

=  *[&]. 

Hence  {a,  ab}'=n[b]. 

To  prove  the  theorem,  we  observe  that,  since  (1)  is  a  divisor  of  a,  and 
a  is  a  divisor  of  ab,  we  have  by  §  2,  iv 

{(i),  ab}  =  {(i),  a}{a,  ab} 

and  hence  n[ab]  =n[a]n[b]. 

We  have  seen  (Chap.  XI,  Th.  5)  that  the  product  of  an  ideal, 
a,  by  its  conjugate,  a',  is  a  rational  principal  ideal  (a).  We  shall 
now  show  that 

n[a]  =  \a\; 
or  in  other  words, 

Theorem  4.  //  a  be  an  ideal  of  a  quadratic  realm  and  a'  its 
conjugate,  then 

aa'  =  (n[a]). 

We  have  act'  =  (a)  (Chap.  XI,  Th.  5),  where  a  is  a  rational 
integer  which  may  be  assumed  to  be  positive. 

Hence  n[a]n[a']  =n[(a)]  *=a2  (Th.  i.Cor.2). 


CONGRUENCES    WHOSE    MODULI    ARE   IDEALS.  337 

But  n[a']  =n[a]  (Th.  i,Cor.  i). 

Hence  n[a]=a, 

and  aa'=0[a]). 

This  theorem  for  the  general  realm  of  the  wth  degree  is  that 
aa'a"  •••  a*"-1)  =  0[a]),  where  a',  a",  ..^at"-1)  are  the  conjugates  of  a. 
The  proof  in  the  case  of  the  quadratic  realm  here  given  is  much  simplified 
by  having  seen  (Chap.  XI,  Th.  5)  that  in  a  quadratic  realm  the  multipli- 
cation of  a  by  a'  gives  a  principal  ideal.     See  Hilbert :  Bericht,  p.  191. 

This  property  of  the  norm  of  an  ideal  might  be  taken  as  its 
definition.  It  would  then  be  exactly  in  line  with  that  of  the 
norm  of  an  integer.  From  Th.  4  it  is  evident  that  n[a]  is  divis- 
ible by  a,  as  in  the  case  of  integers. 

Theorem  5.  The  norm  of  a  principal  ideal,  (a),  is  equal  to 
the  absolute  value  of  the  norm  of  the  integer  a  defining  the  ideal; 
that  is, 

»[(a)]  =  |«[a]|.' 

Let  (a)  be  any  principal  ideal  and  (a')  its  conjugate. 

Then  (a)(a')  =  (»[(a)])(Th.  4), 

and  also  (a)  (a')  =  (aaf). 

But  aa'  =  n[a]=a, 

a  rational  integer,  since  the  norm  of  an  algebraic  integer  is  a 
rational  integer,  and 

n[(a)]=b, 

a  positive  rational  integer. 

Hence  (a)  =  (fc). 

Since  a  is  therefore  divisible  by  b,  and  b  by  a,  we  have 

\a\=b, 
and  hence 

»[(*)]  =  Ma]|, 

'x Hilbert:  Bericht,  Satz  20. 

22 


338  CONGRUENCES   WHOSE   MODULI   ARE   IDEALS. 

Theorem  6.  The  norm  of  a  prime  ideal,  p,  is  a  power  of  the 
rational  prime  which  p  divides.1 

Let  1,  w  be  a  basis  of  the  realm  and  p  =  (a,  b  -j-  c&),  where 
a,  b  -f-  Co)  is  a  canonical  basis  of  p.  It  is  evident  that  a  is  a  prime, 
for,  if  a=axa2,  then  since  p  divides  a,  it  must  divide  either  ax 
or  a2,  say  alt  then  at  would  be  a  number  of  p,  which  would  be 
contrary  to  the  hypothesis  that  a,  b  -\-  Cw  is  a  canonical  basis  of  p, 
and  hence  that  a  is  the  smallest  rational  number  of  p.  Hence  a 
is  a  prime,  p. 

We  have  then 

(P)=pa, 

whence  n[(P)]  =w[^)]«[a], 

and  P*  =  n[p]n[a],  (Th.  1,  Cor.  2). 

Hence,  since  n[£]  and  n[a]  are  positive  rational  integers,  we 
have  either 

n[P]=P,  15) 

or  n[p]=p2;  16) 

we  call  p  a  prime  ideal  of  the  first  or  second  degree  according  as 
15)  or  16)  occurs;   that  is,  the  norm  of  a  prime  ideal,  p,  is  a 
power  of  the  rational  prime  which  p  divides,  and  the  exponent  of 
this  power  is  called  the  degree  of  p. 
For  example: 

and  hence  (3,  1 +V — 5)  is  a  prime  ideal  of  the  first  degree; 
on  the  other  hand, 

n[(2)]=22  =  4, 

and  hence  (2)  is  a  prime  ideal  of  the  second  degree,  both 
(3,  1  -f-V — 5)  and  (2)  having  been  shown  to  be  prime  ideals. 
Cor.  1.  In  a  canonical  basis,  p,  b  -f-  c<n,  of  a  prime  ideal,  p, 
the  coefficient  c  is  1  or  p,  according  as  p  is  of  the  first  or  second 
degree. 

1  This  theorem  holds  for  realms  of  any  degree,  but  the  method  of  proof 
used  here  is  not  applicable  to  those  of  degree  higher  than  the  second. 
See  Hilbert:  Bericht,  Satz  17. 


CONGRUENCES    WHOSE    MODULI    ARE   IDEALS.  339 

Ex.  i.  If  a  and  b  be  two  ideals  and  a  be  prime  to  «[b],  then  n[a]  is 
prime  to  n[b]. 

Ex.  2.  If  J>i,  p2,  ••-,  pn  be  prime  ideals  of  the  first  degree  no  two  of 
which  are  conjugate,  and  whose  norms  are  pi,  p2,  •••,  pn,  show  that  the 
smallest  rational  integer  in  the  product  pip2  •  •  •  pn  is  pip2  •  •  •  pn. 

Ex.  3.  If  the  ideal  a  does  not  contain  the  factor  (p),  where  p  is  a 
rational  prime,  and  n[a]  be  divisible  by  pn  but  not  by  pn+1,  then  a  is  di- 
visible by  pn,  where  n[p]  =  p. 

§  3.  Determination  and  Classification  of  the  Prime  Ideals  of 
a  Quadratic  Realm. 

The  last  theorem  furnishes  us  with  a  method  for  obtaining 
and  classifying  the  prime  ideals  of  any  quadratic  realm,  &(Vra), 
similar  to  that  employed  for  the  prime  numbers  of  &(/),£(  V — 3) 
and  &(V2)-  We  have  seen  that  every  prime  ideal  divides  a 
rational  prime;  hence,  to  obtain  all  prime  ideals  of  k(^/m)  we 
need  only  factor  all  rational  primes  into  their  prime  ideal  factors 
in  k(ym).  If  p  be  a  prime  ideal  and  p  the  rational  prime  which 
p  divides1  (since  ( — p)  =  (p)  we  may  assume  p  positive),  there 
are,  it  has  been  shown,  two  cases  to  be  distinguished.     That  is,  if 

then  P2  =  n[P]n[\], 

and  we  have  either 

i.  n[p]=p  ;     n[\]=p, 

or  ii.  n[p]=p2;     «[j]=i, 

and  hence  j=  (1). 

From  i  it  follows  by  Th.  4  and  the  unique  factorization  theorem 
that 

(p)=ppf;  that  is,  j  =  to'; 
and  from  ii  that 

(P)=9- 

1  That  only  one  rational  prime  can  be  divisible  by  a  prime  ideal  p  is 
evident  from  the  fact  that,  if  two  primes  p  and  q  were  divisible  by  p, 
then  their  rational  greatest  common  divisor  1  would  be  a  number  of  p, 
and  p  would  be  (1). 


340  CONGRUENCES   WHOSE   MODULI   ARE   IDEALS. 

In  i,  (/>)  is  factorable  into  two  conjugate  prime  ideals  of  the 
first  degree. 

In  ii,  (/>)  is  a  prime  ideal  of  the  second  degree. 

We  shall  now  determine  the  relation  which  the  form  of  p 
bears  to  the  occurrence  of  these  cases,  and  shall  see  that  the 
factorization  of  (p)  depends  upon  whether  the  discriminant  of 
the  realm  is  a  quadratic  residue,  a  quadratic  non-residue,  or  a 
multiple  of  p. 

We  shall  show  first  that  the  necessary  and  sufficient  condition 
for  the  factorability  of  (p)  is  that  d  shall  be  a  quadratic  residue 
of  p  or  divisible  by  p,  hence  proving  incidentally  that  the  condi- 
tion for  the  non- factorability  of  (p)  is  that  d  shall  be  a  quadratic 
non-residue  of  p. 

Suppose  that  i  occurs;  that  is, 

{p)=W-  i) 

Since  n[p]  ==p,  there  are  p  incongruent  number  classes  with 
respect  to  p.  We  may  take  as  representatives  of  these  classes  the 
numbers  o,  I,  •••,  p  —  i;  for,  since  p  is  the  smallest  rational 
number  in  p,  the  differences  of  no  two  of  these  numbers  is  a 
number  of  ps  and  they  are  therefore  incongruent  to  each  other 
with  respect  to  p. 

It  is  evident  that  y/m,  which  is  an  integer,  is  congruent  to  one 
of  these  numbers,  say  a,  with  respect  to  J>;  that  is, 

a — \/m  =  o,  mod  p, 

therefore,  since  a+\/ra  is  an  integer  of  k(^tn), 

(a — Vm)  («+Vw)  =a2  —  m==o,  mod  p, 

and  hence,  since  a2  —  m  is  a  rational  number  and  p  the  smallest 
rational  number  in  p, 

a2  —  m  =  o,  mod  p.  2) 

Hence  that  m  shall  be  a  quadratic  residue  of  p  or  divisible  by  p 
is  a  necessary  condition  for  the  factorability  of  (/>). 

We  must  now  distinguish  between  the  two  cases  p=\=2  and 
p==2. 


CONGRUENCES   WHOSE   MODULI   ARE   IDEALS.  34 1 

First  let  p  =j=  2.     It  may  be  shown  that  in  this  case 

a2  —  m  zaa  o,  mod  p, 

is  a  sufficient  as  well  as  necessary  condition  for  the  factorability 
of  (p)  ;  for  from 

a2  —  m=(a—  yra)(a+yra)=o,  mod  p, 

it  follows  (Chap.  XI,  Th.  9)  that,  if  (p)  be  unfactorable,  either 

a — ym  =  o,  mod  (p), 

or  fl-f\/w  =  o,  mod  (/>), 


and  hence  either 


7—      .r-f  1/1///Z 
#  —  ym  =  -         — p 


,—      x  +  y\/m 
or  «  -f  1/ w  = - p 


3) 


where  .ar  and  y  are  either  both  even  or  both  odd,  the  latter  case 
being  possible  only  when  fgeel,  mod  4. 

The  equations  3)  lead  to  the  impossible  equations 

Hence  3)  are  impossible,  and  that  m  shall  be  a  quadratic  residue 
of  p  or  divisible  by  p  is  a  sufficient  as  well  as  necessary  condition 
for  the  factorability  of  (p).  Therefore  that  m  shall  be  a  quad- 
ratic non-residue  of  p  is  a  necessary  and  sufficient  condition  for 
the  non- factorability  of  (p). 

Now  let  the  symbol  (n/q),  where  q  is  an  odd  rational  prime 
and  n  any  rational  integer,  denote  1,  —  1,  or  o,  according  as  n  is 
a  quadratic  residue  or  non-residue  of  q,  or  a  multiple  of  q. 

We  shall  now  obtain  the  factors  of  (p)  when  (p)=pp',  and 
shall  show  that  when  (m/p)  =  1  they  are  different,  and  when 
(m/p)=o  they  are  alike;  that  is,  (p)  is  then  the  square  of  a 
prime  ideal. 


342  CONGRUENCES    WHOSE    MODULI   ARE    IDEALS. 

When  (m/p)  =  i,  a  is  not  divisible  by  p,  and  we  shall  show 
by  actual  multiplication  that 

(P)  =  (P,  a  +V»)  (p,  a—y/m). 
We  have 

(Pi  a-\-\/^n)(p,  a — Vm)=(P2>  Pa — P^/m,  pa  +  PVm>  a2 — m) 

=(p2,  pa —  py/m,  2pa,  a2 —  m) 
=(p2,  pa  —  p\/m,  2pa,  a2  —  m,  p) 

since  p  is  the  greatest  common  divisor  of  p2  and  2pa  and  may 
therefore  be  introduced  into  the  symbol. 
We  shall  show  now  that 

(P,  o+V»i)  +  tt  a—y/m). 
If  they  were  the  same,  both  would  equal 

(Pi  a-\-\/m,  a — -\/m)  =  (p,  a-\-y/m,  2a) 

(f=(fc  a+y/ni,  2a,  i) 
=  (0, 

since  p  and  2a  are  two  rational  numbers  prime  to  each  other  and 
i  may  therefore  be  introduced  into  the  symbol.  Hence  (p)  is 
the  product  of  two  different  conjugate  prime  ideals  when  m  is  a 
quadratic  residue  of  p. 

When  (m/p)  =o,  a  is  divisible  by  p,  and  we  have  by  similar 
analysis 

(P)  =  (P,  V»)  (Ps  —  Vw) 
=  (P,  Vm)2. 

Hence  (p)  is  the  square  of  a  prime  ideal,  when  m  is  divis- 
ible by  p. 

We  see  that,  since  the  discriminant  of  the  realm,  d,  =  m  or  \m, 
according  as  m==  I,  mod  4,  or  =  2  or  3,  mod  4, 

(d/p)  =  (m/p). 

We  may  express  the  results  so  far  obtained  conveniently  as 
follows : 


CONGRUENCES    WHOSE    MODULI    ARE   IDEALS. 


343 


//  p  be  an  odd  rational  prime,  (/>)  is  the  product  of  two  differ- 
ent conjugate  prime  ideals,  or  is  itself  a  prime  ideal,  or  is  the 
square  of  a  prime  ideal,  according  as 

(d/p)  =1,  —  i,  or  o. 

To  obtain  basis  representations  of  p  we  make  use  of  Th.  I,  Cor.  3, 
and  at  once  recognize  that  when  (m/p)  =  i  and  m  =  2  or  3, 
mod  4, 

(P,  «+Vw) 

is  the  required  representation,  for 

p        o 

a  I 

In  the  case  ra=i,  mod  4,  (p,  o+Vw)  is  n°t  a  basis  repre- 
sentation of  p,  for  when  we  express  a  -\- yjm  as  a  linear  combi- 
nation of  the  basis  numbers  1,(1  -\-'\Jm)/2  of  the  realm,  we  have 

that  is  not  a  basis  representation,  since 
/       o" 


a—  1    2 


=  2/**|>]. 


In  this  case  we  can,  however,  get  a  basis  representation  of  p 
as  follows:  since  p  is  odd,  a  can  be  chosen  so  as  to  be  not  only 
a  root  of  a2  =  m,  mod  p,  but  also  odd.  Supposing  this  done,  we 
can  introduce  into  the  symbol  of  p  the  number  (a-\-^/m)/2,  and 
then  omit  a  -f-\/m,  obtaining 

a  +  Vm 


(      a+Vm\ 
=  (A-  — j 

(a  —  1       1  +  |/«*  \ 


which  is  a  basis  representation  of  p,  since 
P       ° 


#—  1 
2 


=  /  =  »[>]. 


344 


CONGRUENCES   WHOSE    MODULI   ARE   IDEALS. 


We  consider  now  the  case  (tn/p)  =  o. 

In  the  cases  m  =  2  or  3,  mod  4,  we  have  as  the  required  basis 
representation 

p=(p,  V«) 
since 


o 

1 


-/-*[»] 


When  m=i,  mod  4,  we  can  introduce  the  number  (p  +y*»)/2 
into  the  symbol  (p%,y/m),  since  p  is  odd,  and  thus  have 

—        /  —  p  +  -j/w 


\        /      p-  1        1  +  i/;;A 


as  a  basis  representation,  since 
1/    o 


=  /  =  *!>]■ 


Let  now  p  =  2. 

We  have  in  all  cases  (w/2)  =  1  or  o;  that  is,  the  necessary 
condition  for  the  factorability  of  (2)  is  always  satisfied.  As  to 
the  sufficiency  of  this  condition  we  must  however  distinguish 
three  cases  according  as  m  =  3,  2  or  1,  mod  4.  When  ms=3, 
mod  4,  we  have  (in/2)  =  1,  and  from  2),  0=1. 

Putting,  therefore,  in  equations  3)  p  =  2  and  a=i,  and  re- 
membering that  when  m  =  3,  mod  4,  #  and  y  must  both  be  even, 
we  see  that  3)  leads  to  the  impossible  equation 

±  I  =  2;r. 

Hence  (m/2)  =  1,  in  the  case  ^  =  3,  mod  4,  is  a  sufficient  condi- 
tion for  the  factorability  of  (2). 
We  have  indeed 

(2)  =  (2, 1  +Vw)  (2, 1  —  y») 

for  2  and    1  +VW  are  evidently  numbers  of  J)  and 
2    o 


CONGRUENCES    WHOSE    MODULI    ARE   IDEALS.  345 

Hence  (2, 1  +  ym)  and  (2,1 — y/ni)  are  the  factors  of  (2). 
But  evidently 

(2,  i+V"w)  =  (2,  1— Vw), 
and  hence 

(2)  =  (2,  1+Vw)2, 

a  result  which  may  be  verified  by  multiplication.     Thus  when 
m==3,  mod  4,  (2)  is  the  square  of  a  prime  ideal. 

When  m  =  2,  mod  4,  we  have  (m/2)  =0,  and  from  2)  a  =  o. 
Putting,  therefore,  in  3)  p  =  2  and  a  =  o,  and  remembering  that 
when  m  =  2,  mod  4,  .r  and  y  must  be  even,  we  see  that  3)  leads 
to  the  impossible  equations 

±  i=2y. 

Hence  (m/2)=o  is  also  a  sufficient  condition  for  the  facto ra- 
bility -of  (2).     We  can  show  just  as  above  that  in  this  case 

(2)  =  (2,  V"02. 

When  mm,  mod  4,  we  have  (m/2)  =  1,  and  from  2)  o=I. 
Putting  £  =  2  and  a=i  in  3)  we  see,  however,  that  x=i, 
y  =  —  1  satisfy  the  first  of  these  equations  and  x=i,  y  =  1  the 
second,  (1 — s/m)/2  and  (i-j-\/w)/2  both  being  integers  of 
k(^m),  when  m=i,  mod  4.  Hence  both  (1 — ^m)  and 
(i-|-yra)  are  divisible  by  (2)  and  nothing  is  known  as  to 
whether  (2)  is  prime  or  not. 

To  determine  when  (2)=pp'  we  may  proceed  as  follows: 
If  (2)  =  pp',  then  o,  1  is  a  complete  system  of  incongruent  num- 
bers with  respect  to  p,  and  hence  (1  -\-\/m)/2  must  be  con- 
gruent to  either  o  or  1  with  respect  to  p;  that  is,  we  must  have 
either 

1  +  ^™  A  c 
a  o,  mod  £, 

1  -I-  i/»i       1  —  Vm 
or  1 = =  o,  mod  to ; 

2  2 

and  hence  in  any  case 


34^  CONGRUENCES    WHOSE    MODULI    ARE   IDEALS. 

But  (1  — m)/4  is  a  rational  integer  and  we  must  have  therefore 
1  —  m 


=  o,  mod  2,  4) 

since  2  is  the  smallest  rational  number  in  p. 
From  4)  it  follows  that 

1  —  m  =  o,  mod  8 ; 

that  is,  rn=i,  mod  8, 

is  a  necessary  condition  for  the  factorability  of  (2)  when  w=i, 
mod  4. 

We  must  now  distinguish  two  cases  according  as  m=i  or  5, 
mod  8.  In  the  latter  case  (2)  is  evidently  a  prime  ideal,  for  4) 
is  no  longer  satisfied.  We  shall  proceed  to  show  that  when 
ra=i,  mod  8,  (2)  is  the  product  of  two  different  conjugate 
prime  ideals.  If  (2)  be  factorable,  p  must  contain  one  of  the 
numbers  (1  -\-\Zm)/2,  (1 — \/m)/2,  and  hence  p'  the  other. 
Moreover,  we  have 

||  2         Oil 

Ilo       l\\ 
Hence,  if  (2)  be  factorable,  we  have 

/      1  -f  Vm\  (      1  —  \/m\ 

(2)  =  ^,  -__j|v__r_j, 

and  this  may  be  shown  to  be  correct,  for  by  multiplication  we  get 

/      1  +  Vm\  (      1  —  Vm\     (  .—  .—    1  —  m\ 

\2%     -j—  J  (s,  -  -^—  j  =  (^4,  1  -  Vmt  1  +  •*,  — -  J 

=  U,  1  -  i/«,  -      -,  2  J 

_=(2), 

since  (1  —  m)/4  and  1 — s/m  are  divisible  by  2,  when  m=i, 
mod  8.     Moreover, 

/      1  +  Vm\    ,    /      I  —  l/w\ 


CONGRUENCES    WHOSE    MODULI    ARE    IDEALS.  347 

for,  if  they  were  the  same,  they  would  both  equal 

(1  +  Vm     1  —  \/m  \       (      1  4-  v  m     I  —  Vm,       \ 

which  is,  of  course,  impossible.     Hence,  when  m=  i,  mod  8,  (2) 
is  the  product  of  two  different  conjugate  prime  ideals. 
We  may  collect  the  results  obtained  for  (2)  as  follows: 
(2)  is  the  square  of  a  prime  ideal  when  m  =  3  or  2,  mod  4; 
it  is  the  product  of  two  different  conjugate  prime  ideals,  when 
m  ==  1,  mod  8,  and  it  is  a  prime  ideal  when  m  =  5,  mod  8. 

We  have  evidently  as  basis  representations  of  the  factors  of 
(2)  in  these  cases  respectively 

(2)  =  (2,  1  +\/m)2,  (2)  =  (2,  ym)\ 

I  +  Vm  \  (      1  —  Vm 


,  .        /      1  +  Vm\  (      1  —  Vm\ 

(2)  =  (2,  1  -fVw)- 


Let  now  the  symbol  (w/2)  denote  1,  —  1,  or  o  according  as  n  is 
a  quadratic  residue  or  non-residue  of  8  or  is  divisible  by  2,  and 
observe  that,  when  m  =  3  or  2,  mod  4,  d  =  4m,  and  hence  is 
always  divisible  by  2,  and  that  when  m=i,  mod  4,  d  =  m,  and 
hence  is  a  quadratic  residue  of  8  when  and  only  when  w=i, 
mod  8,  and  a  quadratic  non-residue  of  8  when  and  only  when 
m=5,  mod  8.  We  may  now  combine  the  results  obtained  for 
p  =  2  with  those  for  p  =4=  2  in  the  following  theorem : 

Theorem  7.  //  p  be  any  rational  prime,  (p)  is  the  product 
of  two  different  conjugate  prime  ideals  of  the  first  degree,  a 
prime  ideal  of  the  second  degree,  or  the  square  of  a  prime  ideal 
of  the  first  degree,  according  as  (d/p)  =  J,  — /,  or  o.1 

An  ideal  a  of  a  quadratic  realm  such  that  a  =  a'  and  which  con- 
tains as  a  factor  no  ideal  (a),  zvhere  a  is  a  rational  integer  differ- 
ent from  ±  1,  is  called  an  ambiguous  ideal.  The  ambiguous  prime 
ideals  of  a  quadratic  realm  are  evidently  the  prime  factors  of  (d). 

The  following  table  gives  basis  representations  of  the  prime 
factors  of  (p)  in  a  convenient  form  for  reference. 

xSee  Hilbert:  Bericht,  Satz  97. 


348 


CONGRUENCES   WHOSE   MODULI   ARE   IDEALS. 


In  it  a  satisfies  the  congruence  a2  =  m,  mod  p,  and  is,  more- 
over an  odd  integer  in  the  case  when  w=  i,  mod  4. 


©-■ 

(,')- 


m  =  I,  mod  4 


m  hi  2  or  3,  mod  4 


Ex.  1. 


*(V-i3) 


We  have  — 13  =3,  mod  4,  whence  1,  V — 13  is  a  basis  of  &(V — 13) 
and  d  =  —  52. 
Since 

and  i2  as  —  13,  mod  2,  we  have  (2)  =  (2,  1  +  V  —  13) 2.    Since 

(3)  is  a  prime  ideal.     Since 

(5)  is  a  prime  ideal. 

Ex.  2.  Find  basis  representations  of  the  prime  ideal  factors  of  all 
rational  primes  less  than  20  in  the  realms  fc(V —  7),  fc(Vn)  and 
*(V30~). 

Ex.  3.  If  the  norm  of  any  ideal  be  divisible  by  an  odd  power  of  a 
rational  prime,  p,  then  p  is  factorable  into  two  conjugate  prime  ideals 
of  the  first  degree. 

§  4.    Resolution  of  any  Given  Ideal  into  its  Prime  Factors. 

We  have  in  the  last  section  given  a  general  method  for  resolv- 
ing any  principal  ideal  defined  by  a  rational  prime  number  into 
its  prime  ideal  factors. 

The  resolution  of  any  given  ideal  a  can  be  effected  by  observ- 
ing that  the  product  of  the  norms  of  the  prime  factors  of  a  must 
equal  n[a],  and  hence  the  only  possible  prime  factors  of  a  are 
the  prime  ideal  factors  of  the  rational  primes  which  divide  n[a]. 


CONGRUENCES    WHOSE    MODULI    ARE    IDEALS.  349 

We  then  determine  by  actual  multiplication  which  of  the  finite  num- 
ber of  prime  ideals  satisfying  this  condition  are  the  proper  ones. 

We  shall  see  that  the  resolution  of  any  ideal  a,=  (alf  a2,  •  • • ,  a„) , 
can  be  made  to  depend  upon  the  resolution  of  the  principal  ideals 
(aj,  (a2),  •••,  (a„),  and  shall  illustrate  by  the  following  ex- 
ample the  resolution  of  a  principal  ideal  into  its  prime  factors. 

Let  &(V — 5)  be  the  given  realm  and  (io+V — 5)  be  the 
given  ideal;  then 

n[(io+Vzr5)]  =  io5  =  3-5-7- 


Hence  (io-f-V — 5)  must  be  the  product  of  three  prime  ideals 
whose  norms  are  respectively  3,  5  and  7.  The  prime  ideals  whose 
norms  are  3  are  evidently  (3, 1  -f~V — 5)  and  (3, 1  — V — 5)-  The 
only  one  whose  norm  is  5  is  (V — 5)-  Those  whose  norms  are 
7  are  (7,  3  +VZI5)  and  (7,  3—  V^S). 

By  multiplication  we  can  determine  which  of  the  four  possible 
combinations  of  these  ideals  is  the  correct  one.  We  can,  however, 
materially  shorten  the  process  by  observing  that,  if  (10+V — 5) 
be  divisible  by  (7,  3 — V — 5)> tnen  (I0+V — 5)  *s  a  number  of 
(7,  3—  V:zr5);  that  is, 

(7,  3—  V=5j=(7,  3—  V^5,  io  +  V^) 

=  (7,  3—  V=5,  io+V=5,  13) 
=  (7,  3—  V^  io+V^,  13,  1) 

=  (1), 

which  is  impossible. 

Hence  (7,  3 — V — 5)  is  not  a  factor  of  (10+V — 5)- 
Since  7,  3 — V — 5  is  a  Dasis  °*  (7>  3 — V — 5)  we  could  have 

determined    whether   or   not    10 +V — 5    is   a    number   of    (7, 

3 — V — 5)  by  seeing  whether  or  not 

io+V^5  =  7-r+  (3—  Vz:r5)y 
where    x    and    y    are    rational    integers.      This    equation   gives 
.a- =13/7,  yz= — 1,  and  it  is  again  proved  that  (10 +V — 5)  is 
not  divisible  by  (7,  3 — V — 5)-     In  like  manner  we  can  show 
that  (3,  1 — V — 5)  does  not  divide  (10+V — 5)-     Hence 
(io+V=5)  =  (3>  i+V=5)(V=5)(7,  3+V-"5). 


350  CONGRUENCES   WHOSE    MODULI   ARE   IDEALS. 

Had  we  first  tested  either  (7,  3+V — 5)  or  (3>  I  +V — 5) 
we  should  have  found,  of  course,  that  (10 +V — 5)  was  divis- 
ible by  it. 

If  n[(a)]  be  divisible  by  a  higher  power,  pr,  than  the  first  of  a 
rational  prime,  p,  then  either  (p)  is  a  prime  ideal  in  which  case  a 
is  divisible  by  pr/2,  this  case  being  possible  therefore  only  when 
r  is  even,  or  (p)  is  the  product  of  two  conjugate  prime  ideals, 
p,  p',  of  the  first  degree. 

In  this  case  (a)  may  be  divisible  by  both  p  and  p',  and  hence 
a  by  p,  or  (a)  may  be  divisible  simply  by  a  power  of  one  of  the 
ideals,  say  p. 

If  a  =  peax, 

where  ax  is  not  divisible  by  p,  then  (ax)  cannot  be  divisible  by  the 
product  pp'  and  hence,  if  w[(«i)]  be  divisible  by  p8,  then  ax  is 
divisible  by  either  ps  or  p'8,  these  cases  occurring  respectively  as 
(ax)  is  divisible  by  p  or  p'. 

The  resolution  of  any  principal  ideal  into  its  prime  factors  can 
therefore  be  effected. 

Let  now  ct=  (alf  a2,  •  •  •,  an)  be  any  ideal.  Since  a  is  the  great- 
est common  divisor  of  the  principal  ideals  (fltj.),  (a2),  "*>  (a»)> 
we  can  effect  the  resolution  of  a  into  its  prime  ideal  factors  by 
resolving  the  ideals  (aj,  (a2),  •••,  (a„)  into  their  prime  factors 
and  taking  their  greatest  common  divisor;  this  will  be  a. 

Ex.  1.  Let  (21,  10  +  V  —  5)  be  the  given  ideal.  We  have  found 
above  that 

Oo  +  v=5)  =  (3,  i  +  V^KV^K;,  3  +  V=5), 

and  we  have  evidently 

(21)  =  (3,  i  +  vT-5)(3,  i— yf^s)(at  a+V^X?.  3— '•^s). 

Hence 

(ai,  10  +  V^=:5)  =  (3,  i  +  V"^r5)(7,  3  +  V"::r5) 

is  the  resolution  of   (21,  10  +  V  —  5)  into  its  prime  factors. 

Ex.  2.  Resolve  the  ideal  (30)  into  its  prime  ideal  factors  in  the  realms 
KV^S),  Hy/^7)  and  *(V30)._ 

Ex.  3.  Resolve  the  ideal  (24  —  V2^)  into  its  prime  ideal  factors  in  the 
realm  k{\/26). 

Results  should  be  verified  by  multiplication. 


CONGRUENCES    WHOSE    MODULI    ARE    IDEALS. 


351 


There   are   many   devices   which   shorten   numerical   work   with   ideals, 
some  of  which  will  be  illustrated  later  in  the  solution  of  examples. 

§  5.     Determination  of  the  Norm  of  any  Given  Ideal. 

If  an  ideal  has  been  resolved  into  its  prime  factors,  or  if  we 
have  a  basis  of  the  ideal,  its  norm  is  easily  found. 
Let  a,=  (alt a2,  •  •  -,a„),  be  the  given  ideal,  and  let 

be  the  resolution  of  a  into  its  prime  factors ;  then 

w[a]=«[PiMl>2]  --n[pr]. 

If  we  have  a  basis  a^  +  a2w2,  b1oil  -\-  b2w2  of  0,  we  have,  of 
course,  at  once 


*[tt]  = 


Theorem  8.  The  greatest  common  divisor  of  the  norms  of 
the  numbers  of  a  is  n[a]. 

Let  n[a]  —a,  and  let  a  be  a  number  of  a  such  that  {a) /a  is 
prime  to  (a).  Then,  if  a'  be  the  conjugate  of  a  and  a'  the  con- 
jugate of  q,  we  have  (a?) /a'  also  prime  to  (a),  and  hence 
(n[a])/(a)  prime  to  (a).  Therefore  a  is  the  greatest  common 
divisor  of  n[a]  and  n[a],  and  hence  of  the  norms  of  all  num- 
bers of  a.1 

It  should  be  observed  that  the  greatest  common  divisor  of  the  norms 
of  the  numbers  defining  a  is  not  necessarily  n[a],  though,  of  course, 
n[a]  is  a  divisor  of  it;  for  example, 

(1  +  V~5,  1  -  V31!))  =  (2,  1  +  >/— 5) 
is    an    ideal    whose    norm    is    2,    but    the    greatest    common    divisor    of 
«[i  +  V  —  5]  and  «[i  —  V  —  5]  is  6. 

§  6.    Determination  of  a  Basis  of  any  Given  Ideal. 

Let  a,  =  (alta2,  ••-,«„),  be  the  given  ideal  and  let  11  [a]  be 
known.  If  two  numbers,  ai,  =  a1oix  +  a2o>2,  dj,  =  b1(D1  +  &2w2,  of 
a  be  known,  such  that 

1 


2 


Hilbert:  Bericht,  Satz  21. 


352  CONGRUENCES   WHOSE   MODULI   ARE   IDEALS. 

then  evidently  ai}  CLj  constitute  a  basis  of  a.  If  no  numbers  sat- 
isfying this  condition  be  known,  we  can  determine  a  canonical 
basis,  a,  b  +  Cm,  of  a,  where  a  and  c  may  be  assumed  positive,  as 
follows : 

We  observe  first  that,  if  ax,  bx  +  cxa>  be  a  canonical  basis  of  an 
ideal  a,  and  e  a  rational  integer,  then  axe,  bxe  +  cxe<a  is  a  canonical 
basis  of  the  ideal  a(e).  The  determination  of  a  basis  of  a  can 
therefore  be  reduced  always  to  the  determination  of  a  canonical 
basis  of  an  ideal  which  is  the  product  only  of  prime  ideals  of  the 
first  degree,  no  two  of  which  are  conjugates. 

Having  resolved  a  into  its  prime  factors,  we  collect  all  pairs 
of  conjugate  prime  ideals  of  the  first  degree  and  all  prime  ideals 
of  the  second  degree.  The  product  of  these  factors  will  be  the 
principal  ideal  (e)  where  e  is  a  rational  integer,  and  we  have 

a  =  ax(e), 

where  ax  is  the  product  of  prime  ideals  of  the  first  degree  only, 
no  two  of  which  are  conjugates,  and  whose  norms  are 

Pi,P2,  '",Pm. 

To  find  a  canonical  basis  alf  bx  -f-  cxw  of  ax,  we  observe  that  ax, 
being  the  smallest  rational  integer  divisible  by  ax,  must  be 
P1P2 '  •  •  Pm,  and  furthermore  that,  since 

axcx  =  n[ax]=  pxp2  •  •  •  pm, 

cx  =  i. 

Hence  pxp2  •  •  •  pm,  bx  +  <*>  is  a  canonical  basis  of  ax,  where  bx  is 
to  be  determined.  Since  n[bx-\-ai]  is  a  rational  integer  and  a 
number  of  ax  we  have 

n[bx  +  o)]  =0,  mod  pxp2  •••  pm;  1) 

that  is,  when  o)=\/m,  bx2  —  m==o,  mod  pxp2  . . .  pm,  2) 

and  when 

1  4-  Vm      (2d,  +  i)2  —  m 
»«-— ,  4  -«P,     modAA'--Al-        3) 

It  will  be  easily  seen  that  2)  and  3)  have  solutions  which  fall 


CONGRUENCES    WHOSE    MODULI    ARE   IDEALS.  353 

into  pairs,  b19  —  b1  and  2b  1  +  i,  —  2bx  —  i,  and  that  each  pair  of 
solutions  of  2)  gives  the  numbers 

bx  +  y/m,    —bx  +  y/in, 
and  each  pair  of  solutions  of  3)  the  numbers 

2bx  -f  1  4-  Vtn       —  2bx  —  1  +  V'm 

One  of  the  numbers  so  obtained  must  belong  to  a±  and  can,  of 
course,  always  be  determined  by  resolving  the  numbers  into  their 
prime  factors  and  thus  finding  out  which  is  divisible  by  at.  It 
can,  however,  usually  be  determined  with  much  less  work  from 
the  fact  that  in  determining  which  of  these  numbers  is  divisible 
by  alt  it  is  helpful  to  observe  that,  if  at  be  divisible  by  pr  but  not 
by  pr+1,  where  n[p]  =  p,  and  if  a  be  one  of  the  numbers  satisfying 
1),  and  n[a]  be  divisible  by  pr  but  not  by  pr+1,  a  itself  not  being 
divisible  by  p,  then  if  a  be  divisible  by  p,  it  is  divisible  by  pr. 

The  above  method  for  determining  a  basis  of  an  ideal  a  de- 
pended upon  the  knowledge  of  the  prime  factors  of  a.  We  shall 
now  explain  how  a  basis  may  be  determined  without  this  knowl- 
edge and  without  that  of  n[a],  giving  therefore  incidentally  a 
method  for  finding  n[a].  We  have  seen  that,  if  among  the  prime 
factors  of  a  there  occur  one  or  more  pairs  of  conjugate  ideals,  0 
is  divisible  by  a  principal  ideal  (e),  where  e  is  a  rational  integer. 
Every  number,  a<  +  bid),  is  therefore  a  number  of  (e)  and  hence 
is  divisible  by  e.  Therefore  at  and  fa  must  be  divisible  by  e. 
Conversely,  if  in  every  number,  ai  +  bn»j  of  a.ai  and  b\  be  divis- 
ible by  e,  then  a  is  divisible  by  (e). 

Let  e  be  the  greatest  common  divisor  of  the  coefficients,  a»,  bi, 
in  all  the  numbers  defining  a,  and  let  ai  =  eriy  bi  —  esi.     Then 

where  ax  is  the  product  of  prime  ideals  of  the  first  degree,  no  two 
of  which  are  conjugates.  We  have  seen  that  a  canonical  basis  of 
ax  has  the  form  a,  b  +  w.  Furthermore  a1=  {rx  +  ^w,  •  •  • ,  r„  +  sn«>) 
and  the  greatest  common  divisor  of  rls  •••,r„,^1,  •••,£„  is  1.     By 

23 


354  CONGRUENCES   WHOSE    MODULI   ARE   IDEALS. 

multiplying  each  number,  r<  +  Siio,  defining  ax,  by  a>,  when 
as=sy/m,  and  by  <o — i,  when  w  =  -J(i  -\-y/m),  we  can  intro- 
duce into  the  symbol  the  numbers,  ti  +  r^ ;  that  is,  such  that  the 
coefficient  of  w  is  r*.  Since  the  greatest  common  divisor  of  the 
coefficients,  rv  --,rn,sx,  '-,sn,  of  w  is  I,  we  can  find  rational  in- 
tegers, Mj,  •  •  •,  un,  vlf  •  •  •,  vn,  such  that 

rxux  -) y  rnun  +  S&  H Y  SnVn  =  I, 

and  hence  can  introduce  into  the  symbol  a  number  b  +  w ;  that  is, 
one  in  which  the  coefficient  of  to  is  I.  This  is  evidently  one  of 
the  desired  basis  numbers.  To  find  the  other  number,  a,  we  pro- 
ceed as  follows.  Every  number  in  the  symbol  can  be  expressed  as 
a  linear  combination  of  b-\-o>  and  a  rational  integer;  thus 

r1  +  s1u  =  s1(b  +  o>)-Yr1  —  s1b  =  s1(b  +  <o)  +  c1, 

where  cx  is  a  rational  integer.      We  have  also 

c1  =  r1  +  sx<1>  —  sx(b  +  g>). 

Hence  we  can  introduce  cx  into  the  symbol  and  omit  rx  +  sxw. 
Proceeding  in  this  manner  with  each  of  the  remaining  numbers, 
we  have  finally  in  the  symbol  only  rational  integers  and  b  +  o>. 
Let  a  be  the  greatest  common  divisor  of  these  rational  integers 
and  n[b~Yo)].  Evidently  we  can  introduce  a  into  the  symbol 
and  omit  all  of  the  rational  numbers ;  that  is,  we  have 

ax=(a,  &  +  o>). 

To  show  that  a,  b  +  w  is  a  basis  of  ax,  we  must  show  that  any 
linear  combination  a(ex  -}-  fxa>)  -j-  (b  ~Y  <o)  (e2  ~Y  /2<u)  of  a  and 
b  +  o>,  where  ^  +  A^,  <?2  +  /2W  are  anv  integers  of  the  realm,  is 
expressible  as  a  linear  combination  ax  ~Y  (b-\-ta)y,  where  x  and 
y  are  rational  integers ;  that  is,  we  must  show  that  the  equation 

ax-Y  (b  +  <»)y=^a(ex-Yfxw)  +  (b  +  <o)  (e2  +  /*») 

is  satisfied  by  integral  values  of  .ar  and  y  for  all  integral  values 
°f  ^u  /u  ^2>  A-  Multiplying,  putting  w2  =  m,  or  <o  +  J(w — i), 
according  as  w=\/ra,  or  i(l  ~YVm)>  equating  coefficients  and 
making  use  of  the  fact  that  w[fr  +  w]   is  divisible  by  a,  we  see 


CONGRUENCES   WHOSE    MODULI    ARE    IDEALS.  355 

easily  that  this  condition  is  satisfied.     Hence  a,  b  +  w  is  a  canon- 
ical basis  of  qx. 

It  is  well  to  observe  that,  when  an  ideal  has  the  form  {a,  fc  +  w),  it  does 
not  follow  necessarily  that  a,  b  -f-  «  is  a  basis.  The  necessary  and  suffi- 
cient condition  for  this  is  that  n  [b  -f-  «]  shall  be  divisible  by  a. 

Ex.  i.  Let  a=(2)(ii)(3,  i  +  V~5>2(7,  3+y— 5)  be  the  ideal 
whose  basis  it  is  required  to  determine.     We  have 

and  n[(Xi]  =  63. 

Hence  63,  b  +  V  —  5  is  a  canonical  basis  of  d,  where  b  is  to  be  deter- 
mined by  the  condition 

b  +  V  —  5  —  o,  mod  fe. 
The  condition 

n[&  -f  V  — 5]  ■  o,  mod  63 ; 
that    is, 

b2  +  5  =  o,  mod  63, 
gives 

fc  =  11,  —  11,  25  or  —25, 

and  hence  as  possible  basis  numbers  of  tti 

n  +  \/^5,   —  H-fV^S.   25 -fV"^,   — 25  +  V^5- 
It   is   easily   seen  that    11  +  V — 5   and  — 25  -f  V  —  5   are  not  divisible 
by  (3,  1  +  V  —  5)  and  hence,  of  course,  are  not  divisible  by  cti,  while  of 
the     two     numbers     — 11-fV  —  5     and     25  -f  V  —  5     remaining,     only 
—  11  +  V  —5  is  divisible  by  (7,  3  +  y/~^T$). 

Hence  — 11  +  V  —  5  is  the  number  required,  a  result  easily  verified 
when  we  see  that 

(-  11  +  V~5)  =  (2,  1  +  V^5)  (3,  1  +  V~5)2(7,  3  +  V^S). 

Hence,  63,  — 11  +  V  —  5  is  a  basis  of  a„  and  (1386,  — 242  + 22V  —  5) 
is  a  basis  representation  of  a. 

Ex.  2.  Let  a  =(210,  70  + 70V ^5,  90  +  30\/—: 5,  —  20  +  40\/^5) 
be  the  ideal  whose  basis  it  is  required  to  determine.  Using  the  second 
method,  we  have  e  =  10  and 

a,  =  (21,  7  +  7>/^=r5,  9  +  3V^r5>  —  2  +  4V^r5)- 

We  see  that  we  can  introduce  the  number  10  -j-  V  —  5  and  have  easily 

at—  (21,  63,  21,  42,  10+  V^S)- 

Now  21  is  the  greatest  common  divisor  of  21,  63,  42  and  m[io-{-  -\J — 5], 
=  105,  and  therefore 

a,=  (21,  10+  y/^S), 

where  21,    10 +V  —  5   is   a  canonical  basis.    A  canonical  basis   of  a  is 
evidently  210,  100  +  10 V  —  5. 


356  CONGRUENCES    WHOSE    MODULI    ARE   IDEALS. 

§  7.  Determination  of  a  number  a  of  any  ideal  a  such  that 
(a) /a  is  prime  to  any  given  ideal  m. 

We  have  proved  the  existence  of  such  a  number  and  shall  now 
show  how  it  may  be  determined  in  any  given  case,  this  problem 
being  not  only  of  interest  but  of  considerable  importance  in  the 
solution  of  certain  problems  to  be  given  later.  The  proof  given 
above  of  the  existence  of  a  furnishes  us  with  a  clue  to  a  method 
for  its  determination,  which  we  shall  illustrate  by  some  examples. 
As  is  seen  from  the  above  proof,  the  determination  of  a  in  the 
general  case  is  dependent  only  upon  its  determination  in  the  case 
where  m  is  a  prime  ideal  p. 

If  a,  =  (a1}a2,  •••,am),  be  any  ideal,  then  some  one,  aif  of  the 
numbers  alf  a2,  •  -,am,x  defining  a,  which  are,  of  course,  all  divis- 
ible by  a,  must  be  indivisible  by  Qp ;  for  otherwise,  all  numbers  of 
a  would  belong  to  ap  and  a  be  divisible  by  ap,  which  is  impossible. 
This  number,  a«,  is  the  required  number  a.  We  have,  therefore, 
merely  to  resolve  in  turn  the  numbers  defining  a  into  their  prime 
ideal  factors  until  we  find  one  which  satisfies  the  required  con- 
dition. 

Consider  the  realm  &(V — 5)  and  let 

a=(2i,  io+V=5);    m=(2,  i+V^). 
Resolving  a  into  its  prime  factors,  we  have 

a=(3»  i+V^X^  3+V=r5). 
Proceeding  now  to  resolve  in  turn  the  numbers  defining  a  into 
their  prime  ideal  factors,  we  have  evidently 
(2i)  =  (3)(7)  = 

(3,  i+V^5)(3,  i-V=::5)(7,  3+V=r5)(7,  3— V=S). 
We  see  now  that  the  quotient 

(2i)/a=(3,  1—  V=^)(7,  3/#V:=r5), 

is   prime  to    (2,  1 +V — 5)>   and   hence   21   is   the   number,   a, 
required. 

1  We  can  reduce  these  always  to  two  but  have  chosen  the  more  general 
case  so  as  to  show  that  this  reduction  is  unnecessary. 


CONGRUENCES    WHOSE    MODULI    ARE   IDEALS.  357 

Also,  since 

(io  +  V=7)  =  (3,  i+V=5)(7,  S+V^IHV^). 
The  quotient 

(2i)/a=(3,  i— V^)^,  3—  V^S) 

is  seen  to  be  prime  to  (2,  1 +V — 5)  5  hence  10  -f-V — 5  will 
also  serve  as  a.  We  could  have  seen  at  once  that  either  21  or 
10  -j-V — 5  would  serve  as  the  required  number,  for  they  are  both 
prime  to  (2,  1 +V — 5),  their  norms  being  prime  to  n{2, 
i  +  V — 5)-  If  0  be  a  principal  ideal  (/?)  and  m  any  ideal,  it  is 
evident  that  the  quotient 

(/8)/(;8)  =  (i) 

is  prime  to  m,  and  hence  (3  is  the  number,  a,  required. 

To  illustrate  the  determination  of  a  in  the  general  case,  let 

a=(2i,  io+V^)  and  m=(i5,  5+V^5). 

Resolving  these  ideals  into  their  prime  ideal  factors,  we  have 
as  above 

a=(3,  i+V—  5) (7,  S+V^), 

and  m=(3,  1  —  V— :5)  ( V— 7)* 

the  last  result  being  easily  obtained  by  the  method  employed  in 
the  factorization  of  a,  or  by  simply  observing  that  each  number 
defining  m  is  divisible  by  V —  5- 
We  have  found 

(21)  =  (3,  i+v=5)(3,  i-\/-5)(7,  3+V=~5) 
(7,  3—  V—  5). 

and    (lO+V=T5)==(3,  i+V=10(7,  3+V— DCV^), 
and  it  is  well  to  see  whether  one  of  these  numbers  does  not  fulfil 
'  the  conditions  demanded  of  a,  this  often  being  the  case.     Here 
we  see,  however,  that  neither  of  the  quotients, 

(2i)/a=(3,  1—  V^)(7}  3—  V^), 

or  (io+V::=:5")/a=(V-=T)J 


358  CONGRUENCES   WHOSE   MODULI   ARE   IDEALS. 

is  prime  to  m,  and  therefore  that  neither  of  the  numbers  21  or 
10  -fV — 5  will  serve  as  a.     Hence  we  must  proceed  to  construct 
a  as  in  the  above  proof. 
We  have 

^  =  (3.  i+V:zr5)(7,  z+yzzri)(z,  1—  V^), 
a2=(3>  l+V=5)<7.  3+Vzr5)(V^5), 

and  it  is  at  once  evident  that  21  and  10  +V — 5  will  serve  as  ax 
and  a2  respectively;  for  the  quotient, 

(2i)/ax=(7,  3—V—5) 
is  prime  to  (  V — 5)>  and 

(io+y=~5)/a2=(i) 
is  prime  to  (3,  1—  V^lO- 

Hence  a  =  2i  +  10  +y:35  =  3I  +V^5 

is  the  number  required. 

This  result  is  easily  substantiated  by  factoring   (31 +V — 5) 
into  its  prime  ideal  factors. 
We  have 

*(3J  +Vzr5)  =966  =  2.3.7.23; 
hence  (31  +V — 5)  is  the  product  of  four  ideals  whose  norms  are 
respectively,  2,  3,  7  and  23.     The  quotient,   (31 +V — 5) At,  1S 
therefore  the  product  of  two  ideals  whose  norms  are  respectively 

2  and  23,  and  hence  is  prime  to  m,  whose  factors  have  the  norms 

3  and  5.     We  indeed  see  easily  that 

(3i  +  V=5)  =  (2,  1  +  V=S) (3.  1  +V:=5) (7.  3  +  V=5) 
(23,  8+V=5). 

§  8.    The  ^-Function  for  Ideals. 

By  <£(m),  where  m  is  an)'  ideal,  we  denote  the  number  of 
integers  of  a  complete  residue  system,  mod  m,  which  are  prime  to 
m;  that  is,  the  number  of  integers  in  a  reduced  residue  system, 
mod  m. 


CONGRUENCES    WHOSE    MODULI   ARE   IDEALS.  359 

Thus,  if  m=(3,  i+y^5),  taking  as  a  complete  residue 
system,  mod  (3,  1  +y— 5),  the  numbers  1,  2,  3,  we  see  that  1 
and  2  only  are  prime  to  (3,  1  +  V^),  3  being  divisible  by  it, 
and  hence 

<K3,  i+Vir5)=2; 
that  is, 

*(3>  i+V:=r5)=w(3J  i+V^I)  — 1. 

Likewise,  if  m=  (3)  =  (3,  1  —  V^S)  (3,  I  +V:=:5),  taking 
as  a  complete  residue  system,  mod  (3),  the  numbers  o,  1, 2,  V— -5, 
1  +V~^r5,  2  +  V^J,  2 V=5,  1  +  2Vzri,  2  +  2VzrI,  we  see 
that  1,  2,  V — 5,  2\/ — 5  are  prime  to  (3)  and  hence 

*(3)=4.     -  rO-i)O-j0 

In  particular,  we  have  <f>(i)  =  1. 

Ex.  1.     Determine  0(1  -f-  V  —  5). 
Ex.  2.    Determine  0(13,  5 -f- V  —  J4)- 


Theorem  9.    //  p  be  any  prime  ideal, 

«ew— ot(i-sj5j). 


By  Th.  2  there  are  in  a  complete  system  of  incongruent  num- 
bers, mod  pe,  exactly  M[pe]/«[p]  that  are  divisible  by  p,  and  hence 
n[pe]  — w[pe]/«[p]  that  are  prime  to  pe.     Hence 


«*—[«(« -«&) 


Ex.    We  have 

*(3,  I  — *'=li)»  =  «[(3.  i-V/=M)3j/i_     '    J) 

V        »(3i  i  —  V—  14)/ 
=  27(i-i) 

=^  /    C? 

The  general  expression  for  <£(m),  where  m  is  any  ideal,  could 
be  deduced  by  a  method  very  similar  to  the  one  first  employed  in 
R.  We  shall  make  use,  however,  of  the  second  method  employed 
in  R  (Chap.  Ill,  §  14),  for  this  was  at  once  applicable  in  k(i) 
(Chap.  V,  §  12),  and  we  shall  find  the  same  to  be  true  in  the  case 
of  ideals.     This  method  depends  in  R,  it  will  be  remembered, 


360  CONGRUENCES   WHOSE   MODULI   ARE   IDEALS. 

upon  the  property  of  the  ^-function  that,  if  a  be  prime  to  b,  then 

cf>(ab)=cf>(a).(f)(b). 

To  prove  this  for  ideals  we  begin  by  proving  the  following 
theorem. 

Theorem  10.  //  m  be  the  product  of  the  ideals  alfa2,  •••,(*« 
that  are  prime  each  to  each,  and  alya2,  --,a8  any  integers  of  the 
realm,  there  exist  integers,  w,  such  that 

wesson,  mod  au  w  =  a2,  mod  a2,  •••,  w  =  as,  mod  as,         1) 

and  all  these  integers  are  congruent  each  to  each,  mod  m.1 

This  theorem  is  proved  most  easily  by  a  method  analogous  to 

the  symmetrical  one  employed  for  the  corresponding  theorems  in 

R  and  k(i). 
Let  m  =  ct^  =  ct2b2  =  •  •  •  =  a*b«. 

Then  &,  +  £>,+  ... +bs=(i), 

and   hence   there   exist   in   the   ideals   hlf  B2,  ••-,&«    respectively, 
numbers  px,  p2,  •  •  • ,  p8,  such  that 

Pi+ AH VPs=i  (Chap.  XI,  Th.  8,  Cor.).  2) 

The  number 

a  A  +  a2P2  H h  a»P* 

satisfies  all  of  the  congruences  1).     For  example,  we  have 

a  A  +  a2p2  H h  0L8Ps  ■■  alt  mod  a± ; 

for,  since  b2, B3,  ••-,BS  are  all  divisible  by  alf  the  numbers  P2, p3, 
•-,ps  are  all  divisible  by  alf  and  from  2)  it  follows  that 
P±  =  i,  mod  ax. 

Furthermore,  if  w  be  any  number  satisfying  the  congruences  1), 
we  have  by  multiplying  them  respectively  with  p1}  p2,  •  •  •,  ps, 

w^i^aj/?!,  mod  m, 

w/32  =  a2p2,  mod  m,  3) 


o)p8^asp8,  mod  m. 


1  See    Chap.    Ill,    §  14,    and    Chap.   V,    §  12 ;    also    Dirichlet-Dedekind : 
§  180,  II. 


CONGRUENCES   WHOSE    MODULI    ARE   IDEALS.  36 1 

Adding  together  the  congruences  3),  and  making  use  of  2), 
we  have 

•1  =  aifit  +  a2p2  H \-  a8ps,  mod  m. 

Hence  all  numbers  satisfying  the  congruences  1 )  form  a  single 
number  class,  mod  tn. 

If  we  let  a1,a2,—-,as  run  through  complete  residue  systems 
with  respect  to  the  moduli  aly  a2,  •  •  • ,  a8  respectively,  the  resulting 

^l>i]tt[a2]  -••n[a8]=n[m] 

values  of  to  evidently  form  a  complete  residue  system,  mod  m. 
The  necessary  and  sufficient  condition  for  w  to  be  prime  to  m  is 
that  alf  a2,  •  •  •,  a8  be  prime  respectively  to  the  moduli  alt  a2,  •  •  •,  a« ; 
for,  from  the  congruences  1)  we  see  that  the  necessary  and  suffi- 
cient condition  that  w  be  prime  to  each  one  of  the  factors  alf  a2, 

•  •  •,  qs  of  m  is  that  each  a  be  prime  to  its  a. 

Hence,  when  alt  a2,  -",a8  run  through  reduced  residue  systems, 
moduli  di,  a2,  ---jCts,  respectively,  the  resulting  values  of  w  form 
a  reduced  residue  system,  mod  m.  We  have,  therefore,  at  once 
the  following  theorem: 

Theorem  ii.     //  alf  o2,  ••-,  a8  be  ideals  prime  each  to  each, 

then 

^(^q.,  •••  a.)  =<f>(a1)<f>(a2)  ---<j>(a8). 

We  can  now  obtain  easily  an  expression  for  <£(m)  when  m  is 
any  ideal  whatever. 

Theorem  12.     //  m,  =  p1eip2e2  •  •  •  p/r,  be  any  ideal,  where  p19  p2, 

•  •  •,  pr  are  the  different  prime  factors  of  m,  then 

♦W— M(t-5ga)(i.-^a)  — (»-5Kj)' 

By  Th.  11  we  have 

<f>(m)=<f>(p^)<f>(p2^)  •••<A(Pre0, 
from  which  by  Th.  9  it  follows  that 

^)=4^<i-^])«[V](I-4])--[^](I-4b) 


362  CONGRUENCES   WHOSE   MODULI   ARE   IDEALS. 

Hence  by  Th.  3 

W  =  4«1](i-^j)(I-4j)-(I-^j) 

Ex.  1.    We  have 

(21,  10  +  V^S)  =  (3,  1  +  V^5)  (7,  3  +  V^5) 
and  hence 

0(21,  10  +  V  —  S)  =21(1  —  |)  (1  — })  =  14- 
Ex.  2.    Find 

*(6-hV  — 14)    and  0(189,  77  +  7 V— "i4>. 
Theorem  13.    //  b  be  any  divisor  of  an  ideal  tit,  and  m  =  ttb, 
the  number  of  integers  of  a  complete  residue  system,  mod  ttt, 
which  have  with  ttt  the  greatest  common  divisor  b  is  <£(tt). 

Since  by  §  I,  ix,  if  the  theorem  be  true  for  any  particular 
residue  system,  mod  m,  it  is  true  for  all,  we  may  take  the  system 
used  in  Th.  2.     We  have  shown  that  the  integers 

Bvlf  $v2,  •••,8vn[n],  4) 

where  8  is  a  number  of  b  such  that  (8)/b  is  prime  to  n,  and 
Vi,v2>  •••>i'ni>]  is  a  complete  residue  system,  mod  tt,  comprise  all 
and  only  those  integers  of  a  complete  residue  system,  mod  tn, 
which  are  divisible  by  b.  Hence  the  integers  of  the  complete 
residue  system,  mod  m,  which  have  with  m  the  greatest  common 
divisor  b,  are  those  of  the  system  4)  in  which  the  coefficient  of 
b  is  prime  to  tt,  and  these  are  <£(tt)  in  number. 

Theorem  14.  //  b1? b2,  "-fhn  be  the  different  divisors  of  ttt, 
then 

I>(b,.)  =  «[m] 

Let  bx,  b2,  •••,b„  be  the  different  divisors  of  m,  including  ttt  and 
(1).     Then 

ttt  =  Xtl1'01  =  tlt2b2  =  ••••=  tttnb«. 

Let  Pi>P*>  ••»/*»[«]  5) 

be  a  complete  residue  system,  mod  ttt,  and  separate  these  numbers 


CONGRUENCES    WHOSE    MODULI   ARE    IDEALS.  363 

into  as  many  classes  as  there  are  different  divisors  of  m,  putting 
into  one  class  the  <f>(m^)  numbers  that  have  with  m  the  greatest 
common  divisor  bx  (Th.  13),  into  another,  the  <f>(m2)  numbers 
that  have  with  m  the  greatest  common  divisor  b2,  etc.  It  is  evi- 
dent that  each  of  the  numbers  5)  will.be  in  one  and  but  one  of 
these  classes,  and  hence,  since  they  are  w[m]  in  number, 

*0»i)  +  *(**)  H h*(n»)  =n[mj. 

But  mlt  m2,  •••,mn  are  the  different  divisors  of  m,  though  in  a 
different  order  from  that  of  the  b's.  The  theorem  is  therefore 
proved. 

The  proof  here  given  of  this  theorem  is,  it  will  be  observed,  dependent 
only  upon  Th.  13.  The  property  of  the  0-function  thus  shown  completely 
defines  the  function  and  we  shall  be  able  to  derive  from  it,  as  in  R,  the 
general  expression  for  0(m).  From  the  general  expression  for  0(m)  may 
then  be  obtained  Th.  II.  We  may  also  obtain  Th.  14  from  the  general 
expression  for  0(tn),  as  in  R.    These  two  proofs  are  left  to  the  reader. 

Theorem  15.  //  m  be  any  ideal  other  than  (r),  whose  prime 
factors  are  p15p2,  ••-,£»•,  and  b  any  divisor  of  m  other  than  m, 
and  if  we  separate  all  ideals  of  the  form 

m 


no  p  being  repeated,  into  two  classes,  I  and  II,  putting  in  class  I 
those  such  that  m  is  divided  by  none  or  by  the  product  of  an  even 
number  of  the  p's,  and  in  class  II  those  such  that  m  is  divided  by 
the  product  of  an  odd  number  of  the  p's,  then  exactly  as  many 
ideals  of  the  one  class  are  divisible  by  b  as  of  the  other. 

We  see  that  the  positive  and  negative  terms  of  the  developed 
product1 


in 


(-k)(-s)-(-k) 


coincide  respectively  with  the  ideals  of  classes  I  and  II ;  that  is, 

1  No  meaning  of  addition  or  subtraction  is  to  be  abscribed  to  the  +  or 
—  sign  attached  to  these  terms,  it  being  simply  observed  that  all  the  terms 
in  the  developed  product  are  ideals,  to  some  of  which  the  sign  +  is 
attached  and  to  others  the  sign  — . 


364  CONGRUENCES   WHOSE   MODULI   ARE   IDEALS. 

denoting  by  2,mlt  Stn2  respectively  the  sums1  of  the  ideals  of  these 
classes,  we  have 


m 


(-*)(-*)■••(-*)-*"■-*"  • 


Let  m  =  p1eiJ)2e2---prer. 

We  shall  prove  the  theorem  first  for  the  case  in  which 

ex  =s*3  ==  •  •  •  ss*er  =5 1 ; 

that  is,  m  is  not  divisible  by  a  higher  power  than  the  first  of  any 
prime  ideal. 

Put  Pip2-"Pr  =  a. 

We  have 

K,-0(,--s)*--(,--s)-^-,)*r,)"-^ 

=  2^  —  2a2, 

where  5alf  2a2  have  meanings  corresponding  to  those  of  ^m^,  2m2. 
If  now  b  be  any  divisor  of  a  other  than  a,  the  number  of  ax 
terms  which  are  divisible  by  h  is  exactly  equal  to  the  number  of 
a2  terms  which  are  divisible  by  ft ;  for,  if  we  put 

a  =  &9i92  •*•  9«» 

where  9i,  92>,,*>9«  are  those  prime  factors  of  a  which  do  not 
divide  b,  then  the  a/s  and  a2's,  which  are  divisible  by  &  are 
respectively  the  positive  and  negative  terms  of  the  developed 
product 

KSi—  i)(?2—  0  •'•  (9.—  I)?  6) 

Moreover,  since  &=H°>  there  is  at  least  one  prime  ideal  which 
divides  a  but  not  & ;  that  is,  there  is  at  least  one  g. 

Hence  there  are  always  exactly  as  many  positive  as  negative 
terms  in  the  developed  product  6),  and  consequently  as  many  a/s 

1This  sum  is  to  be  understood  in  a  purely  formal  sense  as  merely  the 
aggregate  of  the  ideals  of  the  class  connected  by  +  signs>  and  has,  of 
course,  no  connection  with  the  notation  for  the  greatest  common  divisor 
given  on  p.  311. 


CONGRUENCES    WHOSE    MODULI    ARE   IDEALS.  365 

as  a2's  divisible  by  B.  The  theorem  is  therefore  proved  when  m 
is  not  divisible  by  a  higher  power  than  the  first  of  any  prime  ideal. 

We  proceed  now  to  prove  the  theorem  for  the  general  case. 

Letting  a,  alf  a2  retain  the  meaning  assigned  above,  we  have 

and  it  is  evident  that  the  ideals  mlf  m2  coincide  respectively  with 
the  products  nat,  tta2.. 

Let  now  b  be  any  divisor  of  m  other  than  m  and  let  g  be  the 
greatest  common  divisor  of  the  two  ideals 

b  =  gb,   and   n  =  gc. 

We  see  that  6  is  a  divisor  of  a,  for  c  is  prime  to  b,  and  ca  is 
divisible  by  b,  since 

ca  _  gca  _  rta  _  m 

~F~~~g¥~  b~~  b"'  " 

and  m  is  divisible  by  b. 

From  7)  it  follows,  since  c  is  prime  to  b,  that,  if  b  =  m,  then 
c=(i)  and  b  =  a.  Conversely,  if  b  =  a,  and  hence  is  divisible 
by  all  prime  factors  of  m,  then  c,  since  it  is  a  divisor  of  m  but 
prime  to  b,  must  be  ( 1 )  and  hence  b  =  tit. 

Excluding  therefore  the  case  b  =  m,  so  that  we  have  always 
b=f=a,  there  are  among  the  ideals  ax  exactly  as  many  that  are 
divisible  by  b  as  there  are  among  the  ideals  a2. 

Since,  moreover,  the  necessary  and  sufficient  condition  that 
an  ideal 

nti  =  ncti  =  gca^ 

or  ttt2  =  tta2  =  gca2, 

shall  be  divisible  by  b,  =  gb,  is  that  a±  or  a2  shall  be  divisible  by  b, 
there  are  exactly  as  many  of  the  ideals  m1  divisible  by  b  as  of  the 
ideals  m2.     The  theorem  is  therefore  proved. 

This  theorem  and  proof  is  interesting  as  illustrating  once  more  how 
exactly  everything  concerning  rational  integers  that  involves  no  property 
other  than  that  of  divisibility,  can  be  carried  over  to  the  general  realm 
in  terms  of  ideals. 


366  CONGRUENCES   WHOSE   MODULI   ARE   IDEALS. 

As  in  the  case  of  rational  integers,  the  following  theorem  can 
be  deduced  from  the  one  just  proved. 

Theorem  16.  a.  If  /(m)  and  F(m)  be  two  functions  of  any 
ideal  m  that  are  connected  by  the  relation 

S/(b)=F(m),  8) 

where  b  runs  through  all  divisors  of  m,  including  m,  then 

/(m)=2F(m1)— ^(m2),  9) 

where  mlf  m2  run  through  the  values  defined  in  the  last  theorem. 
b.     If  f(m)  and  F(m)  be  connected  by  the  relation 

n/(b)=ILF(m),  10) 

then  /(m)  =  TTZ7;    \.  11) 

To  prove  a)  it  is  sufficient  to  observe  that,  if  bx  be  any  divisor 
of  m  other  than  in,  it  is  a  divisor  of  exactly  as  many  of  the  m/s 
as  of  the  m2's  (Th.  15),  and  hence  when  in  9)  we  replace  the  F's 
by  their  values  in  terms  of  the  /'s  from  8),  /(b^  will  occur 
exactly  as  often  with  the  plus  sign  as  with  the  minus  sign.  Hence 
all  terms  in  the  second  member  of  9)  will  cancel  with  the  excep- 
tion of  /(m),  which  occurs  but  once.  The  proof  of  b)  is  similar 
and  will  be  left  to  the  reader. 

From  Th.  16,  a,  we  can  easily  obtain  by  the  aid  of  Th.  14  the 
general  expression  for  <£(m). 

From  Th.  14  we  have 

2</,(b)=4m], 
where  b  runs  through  all  divisors  of  m.     Applying  Th.   16,  a, 
we  have  • 

/(m)=<£(m),    F(m)=n[m], 
and  hence 

<£(m)  =2»[m1]  — 2«[m2]. 
Since,  moreover, 

2ln1-2l„2  =  m(I-i-)(I-l)...(.-l)( 


CONGRUENCES    WHOSE    MODULI   ARE    IDEALS.  367 

and,  if 


m 
m 


then 


[tnj  = 


W>,  •  •  •  Pi 
«[m] 


we  have 


24mi]-24m2]-«[m](I-^;j)(«-^J)...(I-^I) 

and  hence 

^)  =  «[m](I-^])(I-^J)...(I-^]). 

Summing  up  what  has  been  learned  concerning  the  ^-function 
for  ideals,  we  see  that,  exactly  as  in  the  case  of  the  corresponding 
function  in  R,  the  function  possesses  the  two  properties : 

i.  <f>(ah)=<f>(a)'4>(b)  where  a  is  prime  to  b. 

ii.  2<£(b)  =w[m],  where  b  runs  through  all  divisors  of  m;  and 
that  either  one  of  these  properties  completely  defines  the  function, 
and  from  it  may  be  deduced  the  general  expression  for  <£(m)  and 
the  other  properties,  or  we  may  as  in  R  derive  the  general  expres- 
sion for  the  function  directly  from  its  definition,  and  then  from 
it  get  i  and  ii. 

The  conception  of  $- functions  of  higher  order  and  the  theorems 
relating  to  them  which  hold  for  rational  integers  (Chap.  Ill,  §  6) 
can  be  at  once  extended  to  ideals. 

§  9.  Residue  Systems  Formed  by  Multiplying  the  Numbers 
of  a  Given  System  by  an  Integer  Prime  to  the  Modulus. 

Theorem  17.  //  filf  /x2,  »*->/*»£«]  be  a  complete  residue  system, 
mod  m,  and  a  any  integer  prime  to  trt,  then  a^a^,  •••,a/>in[m]  is 
also  a  complete  residue  system,  mod  m. 

The  integers  afi^a^,  •••,«/*«[,„]  are  incongruent  each  to  each, 
mod  m,  for  from 

a.fii  =  a(jLj,  mod  m 


368  CONGRUENCES    WHOSE    MODULI    ARE    IDEALS. 

it  would  follow  that,  since  a  is  prime  to  m, 

fxi^=fij,  mod  m, 

which  is  contrary  to  the  hypothesis  that  filt  /x2,  •••,/An[m]  form  a 
complete  residue  system,  mod  m.  The  integers  a/^,  a/x2,  •  •  • ,  a/An[  ] 
are,  moreover,  n[m]  in  number.  They  form,  therefore,  a  com- 
plete residue  system,  mod  m. 

Cor.  //  p1}p2,  •••,£$(„,)  be  a  reduced  residue  system,  mod  m, 
and  a  be  prime  to  m,  then  ap1}ap2,  •••,OLp<t>(m)  is  also  a  reduced 
residue  system,  mod  m;  for  apx,ap2,  •••,ap^(m)  are  incongrueht 
each  to  each,  mod  m,  prime  to  m,  and  <£(m)  in  number. 

Ex.  Since  1,  2,  3,  V^Ts.  *  +  V^5>  2  +  \/=~S,  W^S,  1  +  zV^, 
2  + 2V  —  5  constitute  a  complete  residue  system,  mod  (3),  and  V  —  5 
is  prime  to  (3),  V^S,  2V==r5,  3\/^5,  —  5,  —  5  +  V^5,  —  5  +  tV^S* 
—  10,  —  10  +  V  —  5,  —  10  +  2V  —  5  is  also  a  complete  residue  system, 
mod  (3). 

Likewise  since  1,  2,  V  —  5,  2.\J  —  5  is  a  reduced  residue  system,  mod 
(3),  V — 5,  2\/ — 5,  — 5,  — 10  is  also  a  reduced  residue  system,  mod  (3). 

If  p  be  any  prime  ideal  and  a  an  integer  prime  to  p,  it  is  evident 
from  the  above  that  there  exists  an  integer  ax  such  that 

0^  =  1,  mod  p. 

We  call  at  the  reciprocal  of  a,  mod  p. 

§  10.    The  Analogue  for  Ideals  of  Fermat's  Theorem. 

The  following  theorem  is  for  ideals  the  exact  analogue  of 
what  Fermat's  Theorem,  as  generalized  by  Euler,  is  for  rational 
integers.  The  similarity  in  the  proofs  of  the  two  theorems  should 
be  noticed. 

Theorem  18.  If  m  be  any  ideal  and  a  any  integer  prime  to 
m,  then 

a*(m)=  1,  mod  m. 

Let  pitPts  •••>?♦(*)  0 

be  a  reduced  residue  system,  mod  m.     Then,  since 

aPl,ap2,  •••,ap<Km)  2) 

is  also  a  reduced  residue  system,  mod  m,  each  number  of  2)  is 


CONGRUENCES    WHOSE    MODULI   ARE   IDEALS.  369 

congruent,  mod  m,  to  some  number  of  I )  ;  that  is, 
aPi       ■  Pji      ] 
aPi      =  Pj-2       Lmodm,  3) 

aP4>{m)   —   Pj^m)  -> 
where  P.  ,    P;  ,     '  '  ',    Qi±, 

are  the  numbers  i),  though  perhaps  in  a  different  order. 
Multiplying  the  congruences  3)  together,  we  have 

«*(w)  -PiP*  '  '  •  P*(«)  -  P/,  fe '  *  *  <W  mod  m' 
from  which,  since  Plp2  --pMm)  is  prime  to  m,  it  follows  that, 
a*(«)ssl>  mod  m. 

Ex.    Let  m  =  (3  +  V*^),  and  a  =  3.     We  see  that   (3)   is  prime  to 
(3  +  V^5)  and  that  0(3  +  V^1-!)  =  6;  whence 
36=i,  mod  (3  +  V^r5)> 
for  36 — 1,  =728,  is  divisible  by  «[(3+ V  —  5)L  =  H,  and  hence  by 

Cor.  1.  If  p  be  a  prime  ideal,  and  a  an  integer  not  divisible 
by  p,  then 

anm_1  =  i,  mod  p. 

This  is  the  exact  analogue  of  Fermat's  Theorem  for  rational 
integers 

Cor.  2.     //  p  be  any  prime  ideal,  and  a  any  integer,  then 

anM=a,  mod  p. 

§  11.    Congruences  of  Condition. 

Just  as  in  the  rational  realm  we  have  so  far  considered  con- 
gruences that  may  be  compared  to  algebraic  identities,  the  values 
of  all  the  quantities  involved  being  given  and  the  congruences 
expressing  simply  the  fact  that  the  difference  of  the  two  num- 
bers is  a  number  of  the  ideal  that  is  the  modulus,  or,  in  other 
words,  this  difference  considered  as  a  principal  ideal  is  divisible 
by  the  modulus. 

We  shall  now,  as  in  the  rational  realm,  consider  congruences 
that  hold  only  when  special  values  are  given  to  certain  of  the 
24 


370  CONGRUENCES   WHOSE    MODULI   ARE   IDEALS. 

quantities ;  that  is,  the  values  of  these  "  unknown  "  quantities  are 
to  be  determined  by  the  condition  imposed  by  the  congruence. 

To  develop  the  theory  of  congruences  of  condition  for  ideal 
moduli  it  is  necessary  to  introduce  the  conception  of  the  con- 
gruence of  two  polynomials  with  respect  to  an  ideal  modulus; 
thus, 

If  f(xlf  x2,  •  -,xn)  be  a  polynomial  in  the  n  undetermined  quan- 
tities xlfx2,  "-,xn  with  coefficients  which  are  integers  of  k(-\/m) 
and  m  be  any  ideal  of  &(\/m),  we  say  that  f(xx,x2,  ••■,xn)  is 
identically  congruent  to  o  with  respect  to  the  modulus  m,  if  all 
its  coefficients  be  divisible  by  m.1 

This  relation  is  expressed  symbolically  by 

f(xx,x2,  >-,xn)  =o,  mod  m. 

Two  polynomials,  f(xt,  x2,  •••,  xn)  and  <j>(xlf  x2,  ..'.,  xn),  are 
said  to  be  identically  congruent  to  each  other,  mod  m,  if  their 
difference  be  identically  congruent  to  o,  mod  m,  or,  what  is  the 
same  thing,  if  the  coefficients  of  corresponding  terms  in  the  two 
polynomials  be  congruent,  mod  m ;  that  is,  in  symbols 

f(x1,x2,---,xn)=<l>(x1,x2,---,xn),  mod  m, 

*/  f(*v  x2,  -  -  •? **)  —  4>  (*u  *2>  •  •  •»  *"»)  ■■  °>  mod  nt. 

For  example ;  we  have 

( i  +  3  V3^)^2  +  $*y  +  7y2  + 1  +  2  v^^ 

(8  +  3V=r5)^2+(2—  V=5)*y  +  2,  mod  (7,  3+V=T). 

If  f(x1,x2,---,xn)^<i>(x1,x2,-'-,xn),  mod  m,  i) 

and  a19a2i  •••fO«  be  any  n  integers  of  the  realm,  then  evidently 

f(a19a2,  •••,an)=</>(a1,a2,  •••,a»),  mod  m.  2) 

If,  however,  1)  does  not  hold,  then  2)  does  not  hold  in  general2 
for  every  set  of  integers  at,a2,  •••9a». 

1  It  will  be  understood  throughout  this  discussion  that  the  coefficients 
of  a  polynomial  are  integers  of  some  certain  quadratic  realm  and  that 
the  modulus  is  an  ideal  of  this  realm. 

2  For  an  exception  see  §  13. 


CONGRUENCES    WHOSE    MODULI    ARE    IDEALS.  37 1 

The  demand  that  #1$xs,  --,xn  shall  have  such  values  and  only- 
such  that  2)  will  hold  is  expressed  by  writing 

f(xlfx2f  ■  -■,xn)==cf>(x1,x2,  ••-,*„),  mod  m.  3) 

Any  set  of  integers  satisfying  2)  is  called  a  solution  of  3). 
The  determination  of  all  such  sets,  or  the  proof  that  none  exists, 
is  called  solving  the  congruence  3).  We  call  3)  a  congruence  of 
condition. 

If  alf  a2,  •  •  • ,  an  and  pl,p2,--m,pn  be  two  sets  of  n  integers     » 
each  and 


a2^p2 
an  =  pn 


,  mod  m,  4) 


then  by  §  1,  v, 

f (<*!,<*» -~, <*n)  =f(P1,P2,~-,pn),  mod  m, 
and  <j>(a1,a2,--,an)=<l>(p1,p2,'-',Pn),  mod  m. 

Hence  if  a1}a2,  •••,#»  be  a  solution  of  3),  px,p2,  '-,Pn  is  also 
a  solution.  Two  solutions  so  related  are,  however,  looked  upon 
as  identical.  In  order  that  two  solutions  be  different  it  is  neces- 
sary and  sufficient  that  the  n  relations  4)  shall  not  hold  simul- 
taneously. 

It  is  evident  from  the  above  that  in  order  to  solve  any  con- 
gruence, as  3),  it  is  sufficient  to  substitute  for  the  unknowns  the 
(w[m]  )n  sets  of  values  obtained  by  putting  for  each  unknown  the 
w[m]  numbers  of  a  complete  residue  system,  mod  m,  and  observe 
which  values  of  f(xx,x2,  •••,«#»)  so  obtained  are  congruent  to  the 
corresponding  values  of  <j>(x1}x2,  "-,xn),  mod  m. 

There  being  only  a  finite  number,  (n[m])n,  of  possible  solu- 
tions, we  can  by  this  process  always  completely  solve  any  given 
congruence. 

If  the  congruence  have  the  form 

f(xlf  x%9  •  •  •,  xn)  ==  o,  mod  m, 

and  alsa2,  •■•,a»bea  solution,  then  f(x±,x2,  --,xn)  is  said  to  be 
zero,  mod  m,  for  these  values  of  x1}x2,  '-',xn. 


372  CONGRUENCES    WHOSE    MODULI    ARE    IDEALS. 

Ex.     The  solutions  of  the  congruence 

(3  +  V^^5)^  +  ^  +  2  =  o,   mod    (3,    i  +  V  —  5), 
are  easily  seen  to  be 

x  =  1,  y=  —  1,      and      x  m  —  1,  y  =  1,  mod  (3,  1  +  V  —  5)  • 

§  12.     Equivalent  Congruences. 

Two  congruences, 

f±(xltxt9  -~,xn)  as fx(x%, #„•••,#»),  mod  m,  i) 

$1(xX9Xt,-"fXn)ma<l»%(x19X2,'",Xn)i  mod  m,  2) 

are  said  to  be  equivalent  when  every  solution  of  the  first  is  a  solu- 
tion of  the  second  and  every  solution  of  the  second  is  a  solution 
of  the  first. 

All  that  is  said  in  Chap.  Ill,  §  10,  regarding  congruences  in 
R  applies  equally  to  congruences  with  ideal  moduli  in  any  realm 
k(  s/m). 

We  have  two  transformations  which  lead  to  equivalent  con- 
gruences; first,  if  1 )  be  the  given  congruence  and 

F1(x1,x2,-'-,xn)^F2(x1,x2,--',xn),  mod  m,  3) 

be  any  identical  congruence,  mod  m,  in  xt,x2,  "-,xn,  we  can  add 
3)  member  by  member  to  1),  obtaining 

AOi, *»  ' ->*%)  +  Fi(*i, *2,  ••*,*•)■■  /2O1, *2,  •  -;xn) 

+  ^2(^,^2,  ••-,*„), mod m, 
a  congruence  equivalent  to  1). 

By  means  of  this  transformation  we  can  transpose  any  term 
with  its  sign  changed,  from  one  member  of  a  congruence  to  the 
other  and  can  thus  reduce  any  congruence,  as  1 ) ,  to  an  equivalent 
congruence  of  the  form 

f{xuxv  ..-,#»)  sso,  mod  m, 

whose  second  member  is  o.  We  shall  hereafter  assume  the  con- 
gruences with  which  we  deal  to  have  been  reduced  to  this  form. 

We  may  also  by  this  transformation  reduce  the  coefficients  of 
f(xlfx2f  ---yXn)  to  their  smallest  possible  absolute  values,  mod  m, 
and  thus  lessen  the  labor  of  solving  the  congruences.     In  partic- 


CONGRUENCES   WHOSE    MODULI   ARE   IDEALS.  373 

ular  we  can  remove  those  terms  whose  coefficients  are  divisible 
by  m.  If  m  be  such  that  a  complete  residue  system,  mod  m,  can 
be  constructed  entirely  of  rational  integers,  all  coefficients  of 
f(xXi  x2,  •  •  •,  xn)  can  be  replaced  by  rational  integers.  Using  then 
this  residue  system  for  substitution  the  work  becomes  greatly 
simplified,  especially  when  we  remember  that  n[a]  divisible  by 
n[m]  is  a  sufficient  as  well  as  necessary  condition  that  a  shall  be 
divisible  by  m,  if  a  be  a  rational  integer. 

Ex.     The  congruence 
(4  +  3V=5)^2  +  (i— V:::5)^+(3  +  7VI=r5)y2+i7  +  4V^5  =  o, 

mod  (7,  3  +  \/=r5),  4) 

is  equivalent  to  the  congruence 

2x2 -f  4*y  +  3/  +  5  =  o,  mod  (7,  3  +  >/—"$). 

This  is  equivalent  to  adding  to  4)  member  by  member  the  identical 
congruence 

(—  2  —  3\/=r5)*2  +  (3  +  \/z:S)*y  —  7V^53>2  —  12  —  4V—  5  =  0, 

mod  (7,  3-f  V^-S), 
—2  —  3V  —  S.  3  +  V-—5i  —  7V^r5,  and  —12  —  4V ^5  being  all 
divisible  by  (7,  3  +  V^r5)- 

A  second  transformation  which  leads  to  an  equivalent  con- 
gruence is  the  multiplication  of  both  members  of  the  congruence 
by  any  integer,  a,  prime  to  the  modulus ;  that  is,  the  congruences 

f(xltx2i  ••-,.*•„)  =0,  mod  m, 

and  af(xlt  x2,  •  •  • ,  xn)  =  o,  mod  m, 

where  a  is  prime  to  m,  are  equivalent. 

Conversely,  we  may  divide  all  the  coefficients  of  a  congruence 
by  any  integer  prime  to  the  modulus,  obtaining  an  equivalent 
congruence 

Ex.     The  congruences 

(3  +  3V^r5)*2  +  9*  —  6  —  i5V^5=o,  mod  (3  +  V"=r5)> 
(i  +  V3I5)*2  +  3*  —  2  —  sV^^o,  mod  (3  +  V^S), 
are  equivalent,  since  (3)  is  prime  to  (3  -f-  V  —  5). 

As  a  special  case  of  the  multiplication  transformation,  as  we 
shall  call  the  second  of  the  above  transformations,  we  have  the 


374  CONGRUENCES   WHOSE    MODULI   ARE   IDEALS. 

multiplication  of  the  congruence  by  —  I ;  that  is,  the  change  of 
sign  of  each  of  its  coefficients. 

§  13.    Congruences  in  One  Unknown  with  Ideal  Moduli. 

The  general  congruence  in  one  unknown  has  the  form 

f(x)  =  a0xn  -f-  axxn~x  -\ -f-  0*£E30,  mod  m,  1 ) 

where  a0,alf  •••,an  are  algebraic  integers  of  any  realm  k,  m  an 
ideal  of  this  realm,  and  n  a  positive  rational  integer. 
If  p  be  an  integer  of  k  such  that 

/(p)  =0,  mod  m, 

p  is  called  a  root  of  1 ) . 

The  same  analogies  that  existed  in  the  rational  realm  in  the  case 
of  congruences  with  one  unknown  when  the  modulus  is  a  prime 
are  easily  seen  to  exist  for  prime  ideal  moduli,  and  their  absence 
in  the  case  of  composite  ideal  moduli  is  equally  marked. 

The  reason  is,  of  course,  that  just  as  in  R  the  product  of  two 
integers  is  divisible  by  a  prime  number  when  and  only  when  one 
of  the  integers  is  divisible  by  the  prime,  so  the  product  of  two 
integers,  that  is,  two  principal  ideals,  is  divisible  by  a  prime  ideal 
when  and  only  when  one  of  the  integers  (that  is,  one  of  the  prin- 
cipal ideals)  is  divisible  by  the  prime  ideal.  Furthermore,  we 
have  the  same  difference  in  the  case  of  congruences  with  prime 
ideal  moduli  between  saying  that  all  the  coefficients  are  divisible 
by  the  modulus  and  that  the  congruence  is  satisfied  by  every 
value  of  the  unknown;  for  example,  as  is  easily  seen  from 
Fermat's  Theorem  as  extended  to  ideals,  the  congruence 

#nm — ^  =  0,  mod  p, 

where  p  is  a  prime  ideal,  is  satisfied  by  every  integer  of  the  realm, 
but  its  coefficients  are  not  all  divisible  by  p. 

Before  taking  up  the  general  congruence  in  one  unknown  with 
ideal  modulus,  we  shall  consider  that  of  the  first  degree.  We 
give  first  two  simple  examples  of  congruences  of  higher  degree. 

Ex.  1.    Let 

(5  +\/=5)x*+  (1  +V-~5)*  +  8  +  3V-5  =  o,mod  (3, 1  +V=5),    2) 


CONGRUENCES    WHOSE    MODULI    ARE    IDEALS.  375 

be  the  given  congruence.     We  observe  first  that 

I+V^SasO,    mod    (3,    i  +  V^5)> 
5  +  V^5  =  i,    mod    (3,    i  +  V^5), 
8  +  3V^~5  =  2,    mod    (3,    i+V^5), 
and  hence  2)  reduces  to 

x2  +  2===o,  mod   (3,  i  +  V^-5)- 
Substituting  the  numbers,  0,   1,  2,  which  constitute   a  complete  residue 
system,  mod  (3,  i  +  V  —  5),  we  have 

2  =  2hj=o,  mod  (3,  I  +  V  —  5). 
1+2  =  3  =  0,  mod  (3,  i  +  V-^)- 
4  +  2  =  6  =  0,  mod  (3,  i  +  V^-5)- 

The  congruence  has  therefore  the  two  roots  1  and  2. 
Ex.  2.     The  congruence 

(5  —  6V^r5)*2  +  7-r+i  =  o,  mod   (1  —  V37^. 

is  equivalent  to  the  congruence 

—  jt-\-x+  1  =  0,  mod  (1  —  V"--5).  3) 

since 

5  —  6\/^~5  =  —  I,  mod  (1  —  V--5)i 

and 

7ael,  mod  (1  —  V^-S). 
Substituting  the  numbers  0,  1,  2,  3,  4,  5,  of  a  complete  residue  system, 
mod  (1  —  V  —  5)»  m  3)»  we  see  that  the  congruence  has  no  roots. 

§  14.    The  General  Congruence  of  First  Degree  with  One  Un- 
known. 

That  there  is  always  one  and  only  one  integer,  £,  of  a  complete 
residue  system,  mod  m,  that  satisfies  the  congruence 

ax  =  /3,  mod  m,  i) 

where  a  and  (3  are  integers,  m  any  ideal  and  a  prime  to  m,  is  evi- 
dent; for,  if  £  run  through  a  complete  residue  system,  mod  m, 
then  one  and  only  one  of  the  resulting  products,  a|i,  is  con- 
gruent to  /?,  mod  m  (Th.  17).  Hence  1)  has  one  and  only  one 
root,  if.  We  proceed  now  with  the  discussion  of  the  general  con- 
gruence of  the  form  1),  removing  the  restriction  a  prime  to  m. 

A  necessary  condition  that  the  congruence  shall  have  a  solution 
is  evidently,   from    (§   I,  ix),  that  (3  shall  be  divisible  by  the 


376  CONGRUENCES    WHOSE    MODULI    ARE    IDEALS. 

greatest  common  divisor,  b,  of  a  and  m.     We  shall  see  that  this 
condition  is,  as  in  the  corresponding  cases  in  R  and  k(i),  also 
sufficient,  and  that,  if  it  be  satisfied,  the  congruence  has  exactly 
n[b]  roots,  incongruent,  mod  tn. 
To  show  this,  let 

m  =  m1b, 

and  take  as  a  complete  residue  system,  mod  m,  the  w[m1]n[b], 
=  w[m],  integers 

rr=i,2,...,n[b] 

where  p  is  a  number  of  m  such  that  (p)/m  is  prime  to  b,  and 

are  complete  residue  systems  with  respect  to  the  moduli  b  and  m1 
respectively. 

We  shall  show  that,  if  (/?)  be  divisible  by  the  greatest  common 
divisor  of  (a)  and  m,  exactly  w[b]  of  the  numbers  2)  satisfy  1). 

Let  pfa-{-  m  De  one  °f  tne  integers  2). 

Since  dp  is  divisible  by  m,  we  have  by  substitution  in  1),  as 
the  necessary  and  sufficient  condition  that  p8h  -f-  /**  shall  satisfy  1 ) , 

ain  =  f$,  mod  m. 

But  since  (a)/b  is  prime  to  m,  the  numbers 

are  all  and  only  those  numbers  of  a  complete  residue  system,  mod 
m,  which  are  divisible  by  b  (Th.  2). 

But  ft  is  divisible  by  b.  Hence  there  is  one  and  only  one  of  the 
integers  3)  to  which  f$  is  congruent,  mod  m. 

Let  this  integer  be  a^i. 

It  is  evident  that  of  the  integers  2) 

pK  + /*i>  p82  + /*i>  •••,p8n[b]  ~\-  IH, 

satisfy  the  congruence  1),  and  are  the  only  ones  that  do  so. 
They  are,  moreover,  »[b]  in  number.  Hence  the  congruence  1) 
has  exactly  n[b]  roots  that  are  incongruent,  mod  m. 


CONGRUENCES    WHOSE    MODULI    ARE   IDEALS.  377 

In  particular,  when  b  =  ( I ) ,  that  is,  when  a  is  prime  to  m, 
the  congruence  has,  as  we  have  already  seen,  one  and  only  one 
root,  all  other  integers  satisfying  it  being  congruent  to  this  single 
one,  mod  m.  In  this  case  by  means  of  Fermat's  Theorem  for 
ideals,  we  can  find,  as  in  the  analogous  case  in  the  rational  realm, 
a  general  expression  for  the  root  of  the  congruence 

a.v==j3,  mod  m,  4) 

where  a  is  prime  to  m,  and  m  is  any  ideal. 
Since  a  is  prime  to  m,  we  have 

a*(m)  ss  1,  mod  m, 
and  hence 

or  a/ta^"0-1  = /?,  mod  m. 

Hence  ^a*(m)_1  is  the  root  of  the  congruence  4). 

The  most  obvious  method  of  solving  any  given  congruence,  and 
one  always  applicable,  is  to  substitute  in  turn  the  numbers  of  a 
complete  residue  system  with  respect  to  the  modulus,  thus  deter- 
mining all  the  roots,  if  any  exist,  or  proving  the  non-existence  of 
a  root.  This  is  usually  the  easiest  method  when  the  norm  of 
the  modulus,  m,  is  small,  and  especially  when  the  numbers 
1,  •••,»[m]  —  1  constitute  a  complete  residue  system,  mod  m. 

This  method  has  already  been  used  in  §  13,  Exs.  1  and  2.  We  shall 
further  illustrate  it  and  also  the  method  depending  on  Fermat's  Theorem 
on  the  congruence 

S^^i  +  V13^  mod   (7,  3  +  VT=r5),  5) 

The  numbers  o,  1,  2,  3,  4,  5,  6  constitute  a  complete  residue  system,  mod 
(7>  3  +  V  —  5),  substituting  them  in  turn,  we  have 

6) 
7) 

8) 

mod  (7,  3  +  V  — 5),      9) 

10) 

11) 

12) 


0—  (i  +  V  —  5)=  —  i  —  V  —  5#eO 
5—  (i  +  \/^~5)  =  4  — V^Ssso 
10— (1  +  V^5)  =  9  —  V^H^o 

15—  (1  +  V—^)  =  14  —  V^^0 

20—  (1  +  V:i35)  =  19  —  V--5  450 
25—  (1  +  y/^-5)  =24  —  V^he^o 
30— (1  + v^s)  =29  —  V^Se^o 


378  CONGRUENCES   WHOSE    MODULI   ARE   IDEALS. 

all  of  which  results,  except  7)  and  12),  follow  at  once  from  the  fact 
that  n[— 1  —  V^5L  =6,  n[g  —  V^l],  =86,  «[i4  —  V^L  =201, 
wfig — V  —  5],  =366,  and  «[20,  —  V  —  5],  =846,  are  none  of  them 
divisible  by  n(7,  3  +  V —  5),  =  7,  and  hence  none  of  the  numbers, 
—  1  —  V^5>  9  — V—l.  14— V--&  19  —  V^S  and  29  —  \/~5  can 
be  divisible  by  (7,  3  +  V  —  5)- 

To  obtain  7),  we  observe  that  ^[4 — V  —  5L  =21,  is  divisible  by 
»(7i  3  +  V  —  5).  and  therefore  4  —  V  —  5  may  be  divisible  by 
(7,  3  +  V  —  5).     This  is  seen  to  be  the  case  since 

7— (3  +  V^S)  =  4  —  V^. 
Hence  1  is  a  root  of  5). 

To  obtain  11),  we  proceed  exactly  as  with  7)  and  find  that  the  condition 
w[24  —  V  —  5]  divisible  by  n(7,  3  +  V  —  5)>  which  is  necessary  in  order 
that  24  —  V  — 5  snaH  De  divisible  by  (7,  3  +  V  —  5),  is  satisfied,  but  that 
the  equation 

7x  +  (3  +  V^-S)?  =  24  —  V^5 

gives  as  values  for  x  and  y 

27 

*=7*  /=— «• 

These  not  being  both  integral,  24  —  V  —  5  is  not  divisible  by  (7,  3+V — 5)- 
This  last  result  could  have  been  obtained  also  by  showing  that 

(7,   3  +  V~=~5,  24-V^5)  =  0). 

This  method  is,  in  general,  if  a  be  any  integer  and  b,  =  (&,  /3,),  any 
ideal,  to  show  that  a  is  not  divisible  by  b,  it  is  sufficient  to  show  that 
the  ideal   (j8lf  )82,  a)   contains  a  rational  integer  smaller  than  any  in  b. 

If  we  had  noticed  originally  that,  since  5= — (V — 5)2>  and  (7,  3+V — 5) 
is  prime  to  V  —  5,  the  congruence  has  one  and  only  one  root,  the  work, 
after  finding  that  1  was  a  root,  would  have  been  unnecessary.  It  was 
given  in  full  to  illustrate  this  most  primitive  but  fundamental  method  of 
solution,  which  is  entirely  independent  of  the  above  discussion. 

We  shall  illustrate  now  upon  the  same  congruence  the  method  de- 
pendent upon  Fermat's  Theorem. 

Since  5  is  prime  to  (7,  3  +  V— ~S),  and  0(7,  3  +  V^17^)  =6,  we  see 
that  (1  + V  — 5)58  is  the  root  of  5).    To  show  that 

(i  +  V"zr5)55=i,  mod  (7,  3  +  Vzr5), 
we  observe  that 

i  +  V^ees  —  2,  mod  (7,  3  +  V^:5), 
and 

5  =  — 2,  mod  (7,  3  +  V^5), 
and  hence 

(i  +  VTr5)5CE=(-2)(-2)5==64=i,   mod    (7,   3  +  V^5). 
The  solution  of  a  congruence  of  the  form  1)  where  a  is  not  prime  to  m 
is  perhaps  most  conveniently  accomplished  by  means  of  the  method  sug- 
gested by  the  general  discussion  of  this  case.     We  shall  illustrate  this 
by  two  examples. 


CONGRUENCES    WHOSE    MODULI    ARE    IDEALS.  379 

Ex.  2. 

2x  =  ~,  mod  (i  +  V^^)- 

The  greatest  common  divisor  of  (2)  and  (i  +  V  —  5)  is  seen  to  be 
(2,  1  +  V  —  5),  that  does  not  divide  (7).  Hence  the  congruence  has 
no  root. 

Ex.  3.  

2.r==i  —  V  —  5,  mod   (i  +  V  — 5).  13) 

Since  (1  —  V  —  5)  is  divisible  by  (2,  1  +  V  —  5),  the  greatest  common 
divisor  of  (2). and  (1 +V — 5),  the  congruence  has  11(2,  1  +V — 5),  =2, 
roots. 

We  have 


(3.1  +  V-5). 


(2,  i  +  V-5) 

Taking  as  a  complete  residue  system,  mod   (3,  i  +  V  —  5),  the  num- 
bers o,  1,  2,  and  substituting  these  numbers  in  13),  we  have 

o—  (1  —  V=S)=—  1+  V^5  +  o  " 
2  —  (1  —  V~5>  =  I  +  V=S»o 

4—  (1  —  <=$)  =  3  +  >T=5  +  o 
We  have  therefore,  in  the  notation  of  the  general  discussion, 
W=  1,  mod  (1+  V — 5). 


p.,  mod  (1  +  V  —  5). 


Since 

(3) 


(3,  I4-V-5 


.  =(3, 


is  prime  to  (1  +  V  —  5),  we  may  take  p  =  3,  and  since  o,  1  constitutes  a 
complete  residue  system,  mod  (2,  i  +  V  —  5),  we  have  as  the  two  roots 
of  13) 

3.0+1  =  1,  and  3-1  +  1=4. 

The  reader  may  verify  these  results,  as  found  in  examples  2  and  3,  by 
direct  substitution  of  the   numbers   of   a  complete   residue   system,   mod 

(i  +  V^5). 

These  two  congruences  (Exs.  2  and  3)  will  serve  as  instructive  ex- 
amples of  the  dependence  of  the  entire  theory  of  algebraic  numbers  upon 
the  unique  factorization  theorem,  and  the  necessity  for  the  introduction 
of  the  ideal. 

In  Ex.  2,  2  and  1  +  V  —  5>  considered  merely  as  integers  of  k(y/ —  5), 
are  prime  to  each  other,  and,  were  it  not  for  the  failure  of  the  unique 
factorization  theorem  in  k(yj — 5),  we  should  expect  the  congruence 
therefore  to  have  a  single  root  in  accordance  with  the  results  obtained 
in  R  and  k(i).  Substituting  the  numbers  of  a  complete  residue  system, 
mod  i  +  V  —  5,  we  find  that  it  has  no  root. 

Likewise  in  Ex.  3,  considering  the  numbers  involved  merely  as  integers 
of  £(V  —  5)>  we  should  expect  the  congruence  to  have  a  single  root. 


380  CONGRUENCES   WHOSE   MODULI   ARE   IDEALS. 

Substituting  the  numbers  of  a  complete  residue  system,  mod  1  +  V  —  5» 
we  find  that  it  has  two  roots.  The  reason  for  these  discrepancies  is  made 
plain  when  we  resolve  2,  7,  1  —  V  —  S»  J  +  V  —  5,  into  their  prime 
ideal  factors. 

§  15.  Divisibility  of  one  Polynomial  by  another  with  re- 
spect to  a  Prime  Ideal  Modulus.  Common  Divisors.  Common 
Multiples. 

If  p  be  any  prime  ideal  of  a  realm  k,  we  have  the  following 
definition : 

A  polynomial,  f(x),  is  said  to  be  divisible  with  respect  to  the 
modulus  p  by  a  polynomial  <f>(x),  when  there  exists  a  polynomial 
Q(x)  such  that 

fi*\mQ(j*)+{x)l  mod  p. 

We  say  that  <f>(x)  and  Q(x)  are  divisors  or  factors,  mod  p,  of 
f(x),  and  that  f(x)  is  a  multiple,  mod  p,  of  <£(•*")  and  Q(x). 
The  sum  of  the  degrees  of  the  factors  of  f(x)  is  evidently  equal 

to  the  degree  of  f(x). 

0 

The  coefficients  of  f(x),  <K#)  and  Q{x)  are  understood  to  be  in- 
tegers of  k. 

Ex.    It  is  easily  seen  that 

(4  +  3V"=r5)^-^  +  ^  +  V=l^+(i  +  V"=l)^  +  2 

=  (V-5^2+(I+V:==r5)^+2)((3+2V=r5)^+i),  mod  (7,  S+V^JO'* 

Hence 

V=r5^2+(i  +  V:=:r5)^  +  2    and     (3  +  2V'=5)*a  +  1 

are  divisors,  mod  (7,  3  +  -y/  —  5),  of 

We  have  the  same  consequences  of  this  definition  and  the  same 
definitions  of  common  divisor  and  common  multiple  for  prime 
ideal  moduli  as  for  rational  prime  numbers  (Chap.  Ill,  §  15). 

§  16.  Unit  and  Associated  Polynomials  with  respect  to  a 
Prime  Ideal  Modulus.    Primary  Polynomials. 

We  see  as  in  the  rational  realm  that  the  integers  of  the  realm, 
not  divisible  by  p,  divide  every  polynomial  with  respect  to  the 


CONGRUENCES   WHOSE   MODULI   ARE   IDEALS.  38  I 

modulus  p,  since  they  divide  i,  mod  p,  and  that  these  are  the  only 
polynomials  having  this  property. 

We  call  therefore  the  integers  of  k,  which  are  not  divisible  by 
p,  the  unit  polynomials,  mod  p,  or  briefly  the  units,  mod  p. 

Since  two  polynomials  that  are  congruent,  mod  p,  are  consid- 
ered as  identical,  we  can  take  as  the  units,  mod  p,  the  integers  of 
any  reduced  residue  system,  mod  p. 

Two  polynomials  which  differ  only  by  a  unit  factor,  mod  p,  are 
called  associated  polynomials  and  are  looked  upon  as  identical  in 
all  questions  of  divisibility,  mod  p. 

Two  polynomials  that  are  associated  with  a  third  polynomial, 
mod  p,  are  associated  with  each  other,  mod  p. 

Two  polynomials  that  are  associated,  mod  p,  are  evidently  of 
the  same  degree  and  each  is  a  divisor,  mod  p,  of  the  other. 

Conversely,  if  two  polynomials  be  each  divisible,  mod  p,  by 
the  other,  they  are  associated,  mod  p. 

Two  polynomials  that  have  no  common  factor,  mod  p,  other 
than  the  units,  are  said  to  be  prime  to  each  other,  mod  p. 

Any  polynomial,  f{x),  has  n(p)  —  I  associates,  mod  p.  Of 
these,  one  and  only  one  has  the  coefficient  of  its  highest  degree  i. 
This  one  is  called  the  primary  associate,  mod  p  of  f(x).  For 
example,  the  six  polynomials 

x*  _|_  2.v  —  3,     2x3  -\-  4X  —  6,     3,r3  -\- 6x  —  2, 

4.r3_L.    x—s,     5*3  +  3-r—  1,     6;r3  +  5*  —  4, 

are  associated,  mod  7,  and  x3  -\-2x  —  3  is  the  primary  one. 

§  17.  Prime  Polynomials  with  respect  to  a  Prime  Ideal 
Modulus.  Determination  of  the  Prime  Polynomials,  mod  p,  of 
any  Given  Degree. 

A  polynomial  that  is  not  a  unit,  mod  p,  and  that  has  no  divisors, 
mod  p,  other  than  its  associates  and  the  units,  is  called  a  prime 
polynomial,  mod  p.  If  it  has  divisors,  mod  p,  other  than  these 
it  is  said  to  be  composite,  mod  p. 

We  can  determine  the  primary  prime  polynomials,  mod  p,  of 
any  given  degree,  n,  by  the  process  employed  in  the  same  case  in 


382  CONGRUENCES   WHOSE    MODULI   ARE   IDEALS. 

the  rational  realm;  that  is,  write  down  all  primary  polynomials, 
mod  p,  of  degree  n;  then,  having  determined  by  multiplying 
together  the  primary  polynomials,  mod  p,  of  degree  less  than  n, 
all  composite  primary  polynomials,  mod  p,  of  degree  n,  we  strike 
them  from  the  list  of  all  primary  polynomials,  mod  p,  of  degree  n. 
Those  left  are  evidently  the  primary  polynomials,  mod  p,  of 
degree  n. 

§  18.  Division  of  one  Polynomial  by  another  with  respect 
to  a  Prime  Ideal  Modulus. 

Theorem  19.  //  f{x)  be  any  polynomial  and  <f>(x)  be  any 
polynomial  not  identically  congruent  to  0,  mod  p,  there  exists  a 
polynomial  Q(x),  such  that  the  polynomial 

f(x)  —  Q(x)<f>(x)==R(x),  mod  p, 

is  of  lower  degree  than  <j>(x). 

The  operation  of  determining  the  polynomials  Q{x)  and  R(x) 
is  called  dividing  f(x)  by  4>(x),  mod  p.  We  call  Q(x)  the  quo- 
tient and  R(x)  the  remainder.  The  proof  of  this  theorem  is  pre- 
cisely the  same  as  that  for  the  corresponding  one  in  the  rational 
realm. 

The  conception  of  the  congruence  of  two  polynomials  with 
respect  to  a  double  modulus  is  the  same  for  a  prime  ideal  as  for 
a  rational  prime  number. 

§  19.  Unique  Factorization  Theorem  for  Polynomials  with 
respect  to  a  Prime  Ideal  Modulus. 

We  shall  now  show  that,  just  as  a  polynomial  whose  coefficients 
are  rational  integers  can  be  resolved  in  one  and  but  one  way  into 
prime  factors  with  respect  to  a  rational  prime  modulus,  so  a 
polynomial,  whose  coefficients  are  integers  of  any  given  quadratic1 
realm,  can  be  resolved  in  one  and  but  one  way  into  prime  factors 
with  respect  to  a  prime  ideal  modulus.  The  proof  will  be  seen 
to  be  identical  with  that  employed  for  rational  numbers.  We 
begin  by  stating  the  following  theorem,  whose  truth  is  evident. 

1  This  holds  for  realms  of  any  degree. 


CONGRUENCES    WHOSE    MODULI    ARE    IDEALS.  383 

Theorem  20.  //  f(x)  ==Q(x)<f>(x)  +  R(x),  mod  p,  every 
polynomial  that  divides,  mod  p,  both  f(x)  and  <f>(x)  divides  both 
tf>(x)  and  R(x)  and  vice  versa;  that  is,  the  common  divisors, 
mod  p,  of  f(x)  and  <f>(x)  are  identical  with  the  common  divisors, 
mod  p,  of  4>{x)  and  R(x). 

Theorem  21.  //  f1(x),  f2(x)  be  any  two  polynomials  and 
p  a  prime  ideal,  there  exists  a  common  divisor  D(x),  mod  p,  of 
fi(x)y  fz(x)>  such  thQt  D(x)  W  divisible,  mod  p,  by  every  com- 
mon divisor,  mod  p,  of  fx(x),  f2(x),  and  there  exist  two  poly- 
nomials, <\>x{x),  <£2(-r)>  such  that 

AWfcW  +/iW*W"^W«  mod  P- 

We  may  evidently  assume  f2(x)  of  degree  not  higher  than 
fx(x).  Dividing  fx(x)  by  f2(x),  mod  p,  we  can  find  two  poly- 
nomials, Q1(x),  fs(x),  such  that 

/,(*) BE &(*)/,(*)  +  /,(*),  mod  p, 

f3(x)  being  of  lower  degree  than  /2(.r). 
Dividing  f2(x)  by  /3(.r),  mod  p,  we  have 

/.(*)■&(*)/,(*)+/,(*),  mod  p, 

where  ft(x)  is  of  lower  degree  than  /3(.r),  and  similarly 
/,(f)  =  0.(*)/«Or)+/.(f)        - 

/„.2(.r)  =  QB.2('.r)/„_1(.v)+/„(^)   -.  mod  p, 

/*.,<*)  «a„(*)/.(*) 

a  chain  of  identical  congruences  in  which  we  must  after  a  finite 
number  of  steps  reach  one  in  which  the  remainder,  fn+1(x),  is  o, 
mod  p,  since  the  degrees  of  that  remainder  continually  decrease. 

By  Th.  20  the  common  divisors,  mod  p,  of  fn(x)  and  fn-^x) 
are  identical  with  those  of  /„_1(^r)  and  fn-2(x),  those  of  /n_i(-^), 
fn-2(x)  with  those  of  fn.2(x),  fn-3(x),  and  finally  those  of  f3(x), 
f2(x)  with  those  of  f2(x),  fx{x). 

But  fn(x)  is  a  common  divisor,  mod  p,  of  fn(x)  and  fn-i(x) 
and   is  evidently  divisible  by  every   common   divisor  of  fn(x) 


384  CONGRUENCES    WHOSE    MODULI    ARE    IDEALS. 

and  fn-i(x)-     Hence  fn(x)  is  the  desired  common  divisor,  D(x), 
mod  p,  of  f±(x)  and  f2(x). 

If  now  we  substitute  the  value  of  f3(x)  in  terms  of  ft(x), 
f2(x),  obtained  from  the  first  of  these  congruences,  in  the  second 
and  the  values  of  f3(x)  and  f4(x)  in  terms  of  ft(x),  f2{x)  in 
the  third  and  continue  this  process  until  the  congruence 

fn-2{x)=Qn_2{x)fn_1{x)  +/»(.*•),  mod  p, 

is  reached,  we  shall  obtain  a  congruence, 

fi(*)+i{*)+M*)+Mi*)mD{s),  mod  p, 

where  faix),  <f>2(x)  are  polynomials. 

Cor.  If  f1(x),  f2(x)  be  two  polynomials  prime  to  each  other, 
mod  p,  there  exist  two  polynomials,  <j>x{x),  <)>2(x),  such  that 

/iO)4>i<»  +/iWfcW.*!i  mod  p. 
In  this  case  D  (x)  is  an  integer,  a,  not  divisible  by  p,  and  we  have 
/iO)$iO)  +f$x)*M(x)"mCh  mod  p, 
whence,  multiplying  by  the  reciprocal  of  a,  mod  p,  we  obtain 

fx{x)4>1{x)+f2{x)4>2{x)  =  i,  mod  p. 

Theorem  22.  //  the  product  of  tzvo  polynomials,  fx  (x),  f2  (x) , 
be  divisible,  mod  p,  by  a  prime  polynomial,  P(x),  at  least  one  of 
the  polynomials  ft(x),  f2{x)  is  divisible,  mod  p,  by  P(x). 

Let  f1(x)f2(x)^Q(x)P(x),modp,  1) 

where  Q(x)  is  a  polynomial,  and  assume  fx(x)  not  divisible,  mod 
p,byP(x). 

Then  fx(x)  and  P(x)  are  prime,  mod  p,  to  each  other  and  by 
Th.  21,  Cor.  there  exist  two  polynomials,  ^(.r),  <f>2(x),  such  that 

/1OH1O)  +P(s)+t{x)mi$  mod  p.  2) 

Multiplying  2)  by  f2{x)  and  making  use  of  1),  we  have 

P(*j[Q(*)+i(*)  +&(*)+*(«)]■/,(*))  mod  fc 

where  Q(x)<}>1(x)  -\-f2(^)<f>2(x)  is  a  polynomial.     Hence  f2(x) 
is  divisible,  mod  p,  by  P(^). 


CONGRUENCES    WHOSE    MODULI    ARE    IDEALS.  385 

Cor.  1.  //  the  product  of  any  number  of  polynomials  be  divis- 
ible, mod  p,  by  a  prime  polynomial,  P{x),  at  least  one  of  the 
polynomials  is  divisible,  mod  p,  by  P(x). 

Cor.  2.  //  neither  of  two  polynomials  be  divisible,  mod  p,  by 
a  prime  polynomial,  P(x),  their  product  is  not  divisible,  mod  p, 
by  P(x). 

Theorem  23.  A  polynomial,  f(x),  can  be  resolved,  mod  p,  in 
one  and  but  one  way  into  a  product  of  prime  polynomials,  mod  p. 

The  proof  of  this  theorem  is  identical  with  the  corresponding 
one  in  the  rational  realm. 

We  can  now  evidently  write  any  polynomial,  f(x),  in  the  form 

fWaaCAW)^?,^))*-  (Pn(*))en,  mod  p, 

where  Px(x),  P2(x),  •••,  Pn(x)  are  the  unassociated  prime  fac- 
tors, mod  p,  of  f(x). 

If  we  take  P1(x),  P2(x),  •••,  Pn(x)  primary,  the  resolution  is* 
absolutely  unique. 

The  representations  of  the  greatest  common  divisor  and  least 
common  multiple,*  mod  p,  of  two  polynomials  are  identical  with 
those  in  the  rational  realm. 

The  resolution  of  any  polynomial  into  its  prime  factors,  mod 
1p,  may  be  effected  by  the  method  employed  in  the  case  of  rational 
numbers. 

§  20.  The  General  Congruence  of  the  nth  Degree  in  One  Un- 
known and  with  Prime  Ideal  Modulus. 

Theorem  24.     If  p  be  a  root  of  the  congruence 

f(x)=a0xn  +  axxn-1-\ f-a„  =  o,  mod  p,  1) 

f(x)  is  divisible,  mod  p,  by  x  —  p,  and  conversely,  if  f{x)   be 
divisible,  mod  p,  by  x  —  p,  p  is  a  root  of  1). 
Dividing,  mod  p,  f(x)  by  x  —  p,  we  have 

f(x)  =  (x  —  P)<p(x)+R(p),  mod  p, 

whence,  since  p  is  a  root  of  1), 

f(x)  =  (x  —  p)<p(x),  mod  p; 

25 


386  CONGRUENCES   WHOSE    MODULI   ARE   IDEALS. 

that  is,  f(x)  is  divisible,  mod  p,  by  x —  p.  The  converse  is 
evident. 

If  f(x)  be  prime,  mod  pf  the  congruence  i)  evidently  has  no 
roots.  The  converse  is,  however,  not  true;  that  is,  f(x)  may  be 
composite,  mod  p,  but  i)  have  no  roots,  for  the  prime  factors 
of  f(x),  mod  p,  may  all  be  of  higher  degree  than  the  first. 

This  theorem  gives  us  another  method  for  determining  the 
factors,  mod  p,  of  the  first  degree  of  any  polynomial  in  x.  Some 
of  these  factors  may  be  alike  and  we  are  led  therefore  to  say 
that  p  is  a  multiple  root  of  order  e  of  i),  if  f(x)  be  divisible,  mod 
p,  by  (x —  p)e  but  not  by  (x  —  p)e+1. 

If,  therefore,  p1,p2,'",pm  be  the  incongruent  roots  of  i)  of 
orders  elt  e2,  . .  • ,  em  respectively,  we  have 

f(x)m(x— pt)H*— p2)*---  (*—f>m)emfi(*),  mod  p, 

where  ft(x)  is  a  polynomial  having  no  linear  factors,  mod  p,  and 
whose  degree  ^  is  such  that 

where  n  is  the  degree  of  f(x). 

Counting  a  multiple  root  of  order  e  as  e  roots,  we  see  that  i) 
has  exactly  as  many  roots  as  f(x)  has  linear  factors,  mod  p,  and 
have  the  following  important  theorem : 

Theorem  25.     The  number  of  roots  of  the  congruence 

f(x)=a0xn-\-a1xn-1-{ \-an  =  o,  mod  p, 

where  p  is  a  prime  ideal,  is  not  greater  than  its  degree. 

Cor.  1.  //  the  number  of  incongruent  roots  of  a  congruence 
with  prime  ideal  modulus  be  greater  than  its  degree,  the  con- 
gruence is  an  identical  one. 

Cor.  2.    //  the  congruence 

f(x)  =0,  mod  p,  2) 

have  exactly  as  many  roots  as  its  degree,  and  <f>(x)  be  a  divisor, 
mod  p,  of  f(x),  then  the  congruence 

<f>(x)  =0,  mod  p, 


CONGRUENCES   WHOSE    MODULI   ARE   IDEALS.  387 

has  exactly  as  many  roots  as  its  degree;   for 

/(*)M+(*)0(*)>  mod  p, 

where  Q(x)  is  a  polynomial  in  x,  and  every  root  of  the  con- 
gruence 2)  is  a  root  of  either  the  congruence 

<f>(x)=o,  mod  p,  3) 

or  of  the  congruence 

Q(x)  =0,  mod  p.  4) 

Moreover,  the  sum  of  the  degrees  of  3)  and  4)  is  equal  to  the 
degree  of  2). 

If,  therefore,  <f>(x)  had  fewer  roots  than  its  degree,  then  Q(x) 
must  have  more  roots  than  its  degree,  which  is  impossible. 

Hence  the  corollary. 

§  21.    The  Congruence  x*(m)  —  1  =  0,  mod  m. 

Although  in  the  case  of  congruences  of  degree  higher  than  the 
first  the  theorem  just  given  tells  all  that  we  can  in  general  say 
regarding  the  number  of  the  roots,  still  there  is,  as  in  the  rational 
realm,  one  important  case  in  which  the  number  of  roots  is  always 
exactly  equal  to  the  degree  of  the  congruence. 

Theorem  26.     The  congruence 

4r*(a)ss  1,  mod  m,  1) 

has  exactly  </>(m)  roots. 

The  4>(m)  integers  of  a  reduced  residue  system,  mod  m,  evi- 
dently satisfy  1).  Moreover,  since  by  §1,  ix  two  integers  con- 
gruent, mod  m,  have  with  m  the  same  greatest  common  divisor 
and  the  greatest  common  divisor  of  (1)  and  m  is  (1),  every  root 
of  1)  must  have  with  m  the  greatest  common  divisor  (1)  ;  that  is, 
be  prime  to  m.  Hence  the  number  of  roots  of  1 )  is  exactly  equal 
to  <£(m),  its  degree. 

Ex.  1.    The  congruence 

jf^+i^BM,  mod(i+1/=5), 

or  *2==  1,  mod  (1  -f  V -~5)> 

has  two  roots,  1  and  5, 
Likewise  the  congruence 

**CW"=*>-  1,  mod  (7,3  +  /=5), 


388  CONGRUENCES    WHOSE    MODULI    ARE    IDEALS. 

or  *6=ee  i,  mod  (/,  3  +  \f=-$), 

has  six  roots,  i,  2,  3,  4,  5,  6. 

Ex.  2.     Consider  the  congruence 

^(2l/^5,  _5+y— 5)  ^  f>  mod  (2^=5,  _  5  +  ^5),  2) 

Since 

(2V"^5,  - 5  +  V-^S)  =  ( V^l)  (2,   1  +  V^), 
we  have 

0(2V^-5>  —  S  +  V^)  =0(V^5)^(2,   1  + V3^)  =4-1  =  4. 
Substituting  therefore  in  the  congruence 

**aasl,  mod  UV— ~S,  —  S  +  V"^). 
the  numbers  o,  1,  2,  3,  4,  5,  6,  7,  8,  9,  which  form  a  complete  residue  system, 
mod  (2V  —  5,  — 5  +  V  —  5)*  we  see  that  the  numbers  1,  3,  7,  9,  which 
form  a  reduced  residue  system,  mod    (2\/  —  5,  — 5  +  V —  5),  are  the 
only  ones  which  satisfy  the  congruence. 

Cor.     If  d  be  a  positive  divisor  of  $(p),  the  congruence 
xa  —  j  ==  o,  mod  p, 

where  p  is  a  prime  ideal,  has  exactly  d  roots. 

This  follows  at  once  from  Th.  25,  Cor.  2,  since  xd — 1  is  a 
divisor,  mod  p,  of  x*M — 1. 

The  congruence  xnV^ — ,r  =  o,  mod  p,  having  the  n [p]  roots 
pu  P2>  * '  •>  pnipi  equal  in  number  to  its  degree,  we  have  the  identical 
congruence 

xnW  —  x==(x  —  Pl)(x  —  P2)  ...  (x  —  PnM),  mod  p. 
For  example 

x7  —  x  =  x(x  —  i)(x  —  2)0  —  3)0  —  4)0—  5)0"  —  6), 
mod  (7,  3+  V^S)- 

§  22.    The  Analogue  for  Ideals  of  Wilson's  Theorem. 

The  result  just  obtained  gives  us  a  proof  of  the  following 
theorem : 

Theorem  27.  If  p  be  a  prime  ideal  and  plf  p2,  •••,  p^^)  a 
reduced  residue  system,  mod  p,  then 

P1P2  •••?,*,(»+ 1=0,  mod  p. 

Since  the  congruence 

X*M  —  1=0,  mod  p, 


CONGRUENCES    WHOSE    MODULI    ARE    IDEALS.  389 

has  exactly  <f>(p)  roots,  Pl,p2,  •  •  -tp^Mt  we  have  by  §  21 


x 


.*(») 


i==(x  —  Pl)(x  —  p2)  •■•  (*  — p*(»),  mod  p. 


Putting  jt  =  o,  we  have 

—  I3s(- pi)(—  p,)  •••  (—  p«»),  mod  p, 

whence,  since  <£(p)  is  even,  except  when  n[p]  =  2, 

P1P2  ~'P4M+  l—°>  mod  p, 
which  evidently  holds  also  when  n[p]  =  2. 

Ex.     Let  p  =  (7,  3  +  V  — 5)  ;    then  1,  2,  3,  4,  5,  6  is  a  reduced  residue 
system,  mod  (7,  3  -f-  V  —  S)j  and  we  have 

1  -2  •  3  .4.  5  -6+1  =  721=0,   mod    (7,   3  +  V— !)• 

§  23.    Common  Roots  of  Two  Congruences. 

The  common  roots  of  two  congruences 

f1(x)==o,  mod  p,  and  /2(.r)=o,  mod  p, 

are  evidently  the  roots  of  the  congruence 

<f>(x)  sago,  mod  p, 

where  <f>(x)   is  the  greatest  common  divisor,  mod  p,  of  ft(x) 
and  f2(x). 

Since  the  congruence 

2*1*1  —  .r==o,  mod  p, 

has  for  its  roots  the  numbers  of  a  complete  residue  system,  mod 
p,  the  incongruent  roots  of  any  congruence 

f(x)  =0,  mod  p, 

will  be  the  roots  of  the  congruence 

<p(x)  =0,  mod  p, 

where  <f>(x)  is  the  greatest  common  divisor,  mod  p,  of  xnlx^  —  x 
and  f(x). 

This  gives  us  another  method  of  determining  all  the  incon- 
gruent roots  of  any  given  congruence  with  prime  modulus. 


390  CONGRUENCES   WHOSE    MODULI   ARE   IDEALS. 

§  24.  Determination  of  the  Multiple  Roots  of  a  Congruence 
with  Prime  Ideal  Modulus. 

The  multiple  roots  of  the  congruence 

/0)=o,  mod  p,  1) 

may  be  determined  just  as  in  the  case  of  rational  integers.  Let 
P(x)  be  a  prime  polynomial,  mod  p,  and  let  f(x)  be  divisible,  mod 
p,  by  [P(x)]e  but  not  by  [P(x)]e+1;  then 

f(x)  =  [P(x)Y<f>(x),  mod  p, 

or,  what  is  the  same  thing, 

where  F(x)   and  <f>(x)   are  polynomials  in  x,  with  coefficients 
which  are  integers  of  the  realm  k,  to  which  p  and  the  coefficients 
of  f{x)  belong,  and  F(x)  is  identically  o,  mod  p. 
Differentiating  2),  we  have 

f(*)«=[P(*)J**[«P(£)#(jr)  +P(,*)*'(*)]  +F'(x), 

where  P'(x),  <f>'(x)  and  F'(x)  are  polynomials  in  x  with  coeffi- 
cients which  are  integers  of  k,  and  F'(x)  is  identically  o,  mod  p, 
for  all  coefficients  of  F(x)  being  divisible  by  p,  all  coefficients  of 
i7'^-)  are  divisible  by  p.     Hence 

f(x)mlP{xy]-1i>x(x),taodp, 

where  $x(x)  is  a  polynomial  in  x,  with  coefficients  which  are 
integers  of  k,  and  is,  moreover,  not  divisible,  mod  p,  by  P(x),  for 

*,(*)  =  «/»(*)+(*)  +P(x)*'(Jr), 

where  P'(x)  is  of  lower  degree  than  P(x)  and  <#>(^)  is  prime, 
mod  p,  to  P(^")-  Therefore  f(x)  is  divisible,  mod  p,  by  the 
prime  factor  P(x)  exactly  once  less  often  than  f(x). 

In  particular,  if  f(x)  be  divisible,  mod  p,  by  (x  —  p)e  but  not 
by  (x  —  p)e+1,  then  f(x)  is  divisible,  mod  p,  by  (x  —  p)e_1  but 
not  by  (x  —  p)e. 

Hence  the  theorem : 


CONGRUENCES    WHOSE    MODULI   ARE   IDEALS.  39 1 

Theorem  28.    If  the  congruence 

f(x)  =0,  mod  p, 

have  a  multiple  root,  p,  of  order  e,  the  congruence 

f(x)  =0,  mod  p, 

has  the  multiple  root  p  of  order  e  —  1. 

If  the  greatest  common  divisor,  mod  p,  of  f(x)  and  f(x)  be 
4>(x),  then  the  roots  of  the  congruence 

<£0)e=o,  mod  p,  3) 

if  it  have  any,  will  be  the  multiple  roots  of  1)  and  each  root  of  3) 
will  occur  once  oftener  as  a  root  of  1)  than  as  a  root  of  3). 

It  may  happen,  of  course,  that  f(x)  and  f{x)  have  a  common 
divisor,  <f>(x),  mod  p,  and  yet  1)  has  no  multiple  roots.  In  this 
case  the  repeated  prime  factors,  mod  p,  of  f(x)  are  of  degree 
higher  than  the  first,  and  <j>(x),  therefore,  contains  no  factor  of 
the  first  degree,  mod  p. 

§  25.  Solution  of  Congruences  in  One  Unknown  and  with 
Composite  Modulus. 

The  solution  of  a  congruence  of  the  form 

f(x)  =  a0xn  +  axxn~x  -| +a„  =  o,  mod  m,  1 ) 

where  m  =  m1m2  •  •  •  mt, 

m19  m2,  ••■•,  mt  being  ideals  prime  each  to  each,  can  be  reduced  to 
the  solution  of  the  series  of  t  congruences 

f(x)  =0,  mod  mlt ' 
f(x)  ^o,  mod  m2, 


f(x)  ^o,  mod  m 


2) 


Every  root  of  I )  is  evidently  a  root  of  each  of  the  congruences 
2),  and  conversely  any  integer,  p,  of  the  realm  which  is  simul- 
taneously a  root  of  each  of  the  congruences  2)  is  a  root  of  1), 
for  if  the  integer  f(p)  be  divisible  by  each  of  the  ideals  mlf  m2, 
•••,  mt,  which  are  prime  each  to  each,  it  is  divisible  by  their 
product. 


392  CONGRUENCES   WHOSE   MODULI   ARE   IDEALS. 

If  therefore  alfa2,---,at  be  roots  of  the  congruences  2)  and 
p  be  chosen  so  that 

p  =  a1}  mod  Bin  ) 

P  =  a2,  mod  m2,  I  3) 

pzz=at,  mod  mt, 
then  p  is  a  root  of  1). 

Since  m1}m2,  ■■■,mt  are  prime  each  to  each,  it  is  by  Th.  10 
always  possible  to  find  p  so  as  to  satisfy  the  conditions  3). 

Let  p1,p2>'">Pt  be  auxiliary  integers  selected  as  in  Th.  10; 
then 

p  =  aj$i  +  a2(32  H h  atfitt  mod  m,  4) 

is  a  root  of  1),  and,  if  the  congruences  2)  have  respectively 
^2>  "->h  incongruent  roots,  then  by  Th.  10  1)  has  lj2  •••  lt  in- 
congruent  roots,  which  are  obtained  by  putting  for  alta2,  •••,&* 
in  4)  respectively  the  llt  l2,  ••-,/*  roots  of  the  congruences  2).  In 
particular,  if  any  one  of  the  congruences  2)  have  no  root,  then 
1)  has  no  root. 

We  may  now  suppose  m  =  p&pf*  •  •  •  prer,  where  the  }>'s  are  different 
prime  ideals,  and  show,  as  in  the  corresponding  case  in  R  (p.  96),  that  the 
solution  of  the  congruence  f(x)  ^o,  mod  pe,  can  be  made  to  depend 
upon  that  of  f(x)  ^o,  mod  p^1,  and  hence  eventually  upon  that  of 
/(jr)^o,  mod  p,  the  same  method  being  applicable  with  slight  modifi- 
cations. 

§  26.    Residues  of  Powers  for  Ideal  Moduli. 

//  a  be  prime  to  the  ideal  m,  and 

fi^a*,  mod  m, 

where  t  is  a  positive  rational  integer,  /?  is  said  to  be  a  power 
residue  of  a  with  respect  to  the  modulus  m. 
For  example,  since 


—  2V—  5=(i+V—  5)3>  mod  (7,  3-f-V— 5), 
we  say  that  — 2\/ — 5  is  a  power  residue  of   1  -|~V — 5>  mod 
(7>  3  +V — 5)-     Two  power  residues  of  a  which  are  congruent, 
mod  m,  to  each  other  and  hence  to  the  same  power  of  a,  are 
looked  upon  as  the  same. 


CONGRUENCES    WHOSE    MODULI    ARE    IDEALS. 


393 


A  system  of  integers  such  that  every  power  residue,  mod  m, 
of  a  is  congruent,  mod  m,  to  one  and  only  one  integer  of  the 
system  is  called  a  complete  system  of  power  residues  of  a,  mod 
m.  These  integers  may  evidently  be  selected  from  among  the 
integers  of  any  reduced  residue  system,  mod  m.  The  following 
table  gives  the  power  residues  of  all  numbers  of  a  reduced  residue 
system,  mod  (7,  3+V — 5),  the  system  taken  being  1,2,3,4,5,6. 


m 

= 

(7, 

3+ V 

0 

a0 

a1 

a2 

a3 

a* 

a5 

a6 

1 

I 

I 

I 

I 

I 

I 

I 

2 

4 

I 

2 

4 

I 

I 

3 

2 

6 

4 

5 

I 

I 

4 

2 

1 

4 

2 

I 

I 

5 

4 

6 

2 

3 

I 

I 

6 

1 

6 

1 

6 

1 

We  ask  now  what  is  the  smallest  value,  ta,  of  t,  greater  than  o, 
for  which 

a*=  1,  mod  m. 

That  such  a  value  of  t  always  exists  and  is  equal  to  or  less  than 
<£(m)  is  evident  from  Th.  10  by  which  we  have,  since  a  is 
prime  to  m, 

a*(m)s=  1,  mod  m. 

Giving  to  ta  the  above  meaning,  we  say  that  the  integer  a  apper- 
tains to  the  exponent  ta  with  respect  to  the  modulus  m. 

We  see,  by  consulting  the  above  table,  that  3  and  5  appertain 
to  the  exponent  6;  that  is,  <£(m),  mod  (7,  3  +V — 5),  that  2  and 
4  appertain  to  the  exponent  3,  mod  (7,  3+V — 5)»  and  that  6 
appertains  to  the  exponent  2,  mod  (7,  3+V — 5). 

It  is  evident  that,  if  a  =  /?,  mod  m,  then  a  and  (3  appertain  to 
the  same  exponent,  mod  m.  Hence  to  find  the  exponents  to 
which  all  integers  appertain,  mod  m,  it  is  only  necessary  to  ex- 
amine the  numbers  of  a  reduced  residue  system,  mod  m. 


394  CONGRUENCES   WHOSE    MODULI   ARE   IDEALS. 

Theorem  29.  If  the  integer  a  appertain  to  the  exponent 
4,  mod  m,  then  the  ta  powers  of  a, 

it,  a,  a8,....,  a*.-1,  1) 

are  incongruent  each  to  each,  mod  m. 

Let  as,  a8+r  be  any  two  of  the  numbers  1). 

If  as+r  =  as,  mod  m,  2) 

then,  since  a  is  prime  to  m, 

ar=i,  mod  m.  3) 

But  r  is  less  than  ta  and  3)  is  therefore  impossible,  since  a 
appertains  to  ta. 

Hence  2)  is  impossible. 

Theorem  30.  //  a  appertain  to  the  exponent  ta,  mod  m,  any 
two  powers  of  a  with  positive  exponents  are  congruent  or  incon- 
gruent, mod  m,  according  as  their  exponents  are  congruent  or 
incongruent,  mod  ta. 

Let  a*1,  a82  be  any  two  powers  of  a,  slf  s2  being  positive  rational 
integers,  and  let 

*i  =  qJa  +  ri»   s*  =  q*ta  +  r*> 

where  qlf  q2  are  positive  rational  integers  and 

o^r1<ta,     o^r2<ta,    rx^r2.  4) 

If  otfi*«-H-i  =  a^-+r2,  mod  m,  5) 

then  ari  =  ar2,  mod  m,  6) 

and  hence,  since  a  is  prime  to  m, 

ari~r2=  1,  mod  m. 
But  from  4)  we  have  o^rx  —  r2  <  tai  whence,  since  a  apper- 
tains to  ta,  mod  m, 

r1  =  r2.  7) 

Therefore  s±  =  s2,  mod  /<.,  8) 

is  a  necessary  condition  that  we  shall  have 

asi  =  a*2j  mod  m.  9) 

Moreover,  from  8)  follow  in  turn  7),  6)  and  5).     Hence  8)  is 
also  a  sufficient  condition  for  the  existence  of  9). 
We  have  therefore 


CONGRUENCES   WHOSE    MODULI    ARE    IDEALS. 


395 


a       E=a'a+1  =  a2'a+1  =  - 


,ta-l 


-mod  m; 


that  is,  the  same  law  of  periodicity  holds  for  power  residues  with 
respect  to  ideal  moduli  as  in  the  case  of  rational  integers. 

This  can  be  verified  by  an  examination  of  the  table  (p.  393), 
where  we  see,  for  example,  that  2  appertains  to  the  exponent  3, 
mod  (7,  3+V^lO,  and  that 
2°  mm  23  sg  26  mm 

mod  (7,  3  +V:=:5), 


2  ==2* 


2-  =  2l 


2'  = 
28  = 


and 


0  =  3  =  6=--. 
1=4  =  7=... 
2^5  =  8=. --J 


mod  3. 


Theorem  31.  The  exponent,  ta,  to  which  an  integer,  a,  apper- 
tains with  respect  to  the  modulus  m,  is  always  a  divisor  of  <£(m). 

Since  a+^ssissa0,  mod  m, 

we  have  by  Th.  30  <£(nt)  =0,  mod  ta. 

Theorem  32.  If  two  integers,  alt  a2,  appertain,  mod  tn,  to  two 
exponents,  tlt  t2,  which  are  prime  to  each  other,  then  their 
product,  a^a2,  appertains,  mod  m,  to  the  exponent,  txt2. 

Let  axa2  appertain  to  the  exponent  t,  then 

(a1o2)#ssi,  mod  m.  10) 

Raising  both  members  of  10)  to  the  ^th  power,  we  have 

a^hta^*  mm  i,  mod  m. 

But  a^^mmi,  mod  m, 

and  hence  a^^i,  mod  tn. 

Therefore,  since  a2  appertains  to  the  exponent  t2,  mod  m,  txt 
must  be  a  multiple  of  t2,  whence,  since  tlf  t2  are  prime  to  each 
other,  it  follows  that  t  is  a  multiple  of  t2. 


396  CONGRUENCES    WHOSE    MODULI   ARE    IDEALS. 

In  like  manner  we  can  show  that  Ms  a  multiple  of  tv 
Therefore,  t  being  a  multiple  of  both  t1  and  t2,  is  a  multiple  of 
their  product,  txt2. 

Hence  the  smallest  possible  value  of  t  for  which  1 )  holds  is  fx#8. 
Therefore,  axa2  appertains  to  the  exponent  ^2,  mod  m. 

Ex.  We  see  from  the  table  (p.  393)  that  2  and  6  appertain,  mod  (7, 
3  +  V  —  5)>  respectively  to  the  exponents  3  and  2,  and  that  their  product, 
12,  ^5,  mod    (7,  3  +  V  —  5),    appertains   to   the   exponent   6,   mod    (7, 

Limiting  ourselves  now  to  the  case  in  which  the  modulus  is  a 
prime  ideal  p,  we  ask  whether  there  are  integers  appertaining  to 
every  positive  divisor  of  <j>(p),  and,  if  so,  how  many? 

An  examination  of  the  table  will  show  us  how  matters  stand 
when  p  =  (7,  3  +V=r5)- 

We  have  <f>(y,  3  -f-V — 5)  =6,  and  the  positive  divisors  of  6 
are  1,  2,  3  and  6. 

To  1  appertains  the  single  integer  I. 

To  2  appertains  the  single  integer  6. 

To  3  appertain  two  integers,  2  and  4. 

To  6  appertain  two  integers,  3  and  5. 

Theorem  33.  To  every  positive  divisor,  t,  of  <f>(p)  there 
appertain  <f>(t)  integers  with  respect  to  the  modulus  p. 

Assume  that  to  every  positive  divisor,  t,  of  <f>(p)  there  apper- 
tains at  least  one  integer,  a.  We  shall  show  that,  if  this  assump- 
tion be  true,  there  appertain  to  t  <f>(t)  integers;  that  is,  to  every 
positive  divisor,  t,  of  <f>(p)  there  appertains  either  ^(t)1  integers 
or  no  integer. 

Let  \J/(t)  denote  the  number  of  integers  appertaining  to  t. 
Each  of  the  integers 

a°=i,a,a2,  •••,a*_1,  11) 

is  a  root  of  the  congruence 

£*■■  1,  mod  p;  12) 

for,  if  ar  be  any  one  of  these  integers,  then 

(ar)t=(at)rBsii  mod  p, 

1  We  consider  t  simply  as  a  rational  integer,  and  <t>{t)  is  to  be  understood 
in  this  sense. 


CONGRUENCES    WHOSE    MODULI    ARE    IDEALS.  397 

since  af  =  I,  mod  p. 

The  integers  n)  are,  moreover,  incongruent  each  to  each,  mod 
p  (Th.  29),  and  being  t  in  number,  are,  therefore,  all  the  roots  of 
12),  since  12)  cannot  have  more  than  t  incongruent  roots  (Th. 
25,  Cor.  2).  But  every  integer  appertaining  to  t  must  evidently 
be  a  root  of  12)  and  we  need  look,  therefore,  only  among  the 
integers  1 1 )  to  find  all  the  integers  belonging  to  t . 

Let  ar  be  as  before  any  one  of  the  integers  11). 

If  ar  appertain  to  t  we  must  have  ar,a2r,  •••,au_1)r  all  incon- 
gruent to  1,  mod  p. 

By  Th.  30  the  necessary  and  sufficient  condition  for  this  is 

ir^o,  mod  t,  13) 

where  i  runs  through  the  values  1,  2,  ••••,   t — 1. 

It  is  easily  seen  that  the  necessary  and  sufficient  condition  that 
13)  shall  hold  is  that  r  shall  be  prime  to  t.  Hence  the  necessary 
and  sufficient  condition  that  any  one  ar  of  the  integers  11)  shall 
appertain  to  t  is  that  its  exponent  r  shall  be  prime  to  t. 

This  condition  is  fulfilled  by  <f>(t)  of  the  integers  11),  and  we 
have  proved  therefore  that 

^(f)  =either$(£)  oro. 

We  shall  now  prove  that  the  latter  case  can  never  occur. 

We  separate  the  <f>(p)  integers  of  a  reduced  residue  system, 
mod  p,  into  classes  according  to  the  divisor  of  <j>(p)  to  which 
they  appertain;  that  is,  if  t1,t2,  --,tn  be  the  positive  divisors  of 
<f>(p)  we  put  in  one  class  the  \p{tx)  integers  of  the  above  system 
that  appertain  to  tx,  in  another  class  the  if/(t2)  integers  that  apper- 
tain to  t2,  etc.  It  is  evident  that  no  integer  can  belong  to  two 
different  classes  and  that  every  integer  of  this  system  must  belong 
to  some  one  of  these  classes. 

The  integers  of  a  reduced  residue  system,  mod  p,  being  <£(p) 
in  number,  we  have,  therefore 

rt*i)+rth) +  "'++(**)  =+(p>* 

But,  considering  <£(p)  simply  as  an  integer  of  R,  we  have  also 
(Chap.  Ill,  Th.  6) 

*('i)  +<K'2)  +  —  +*(*•)  =*(P). 


398  CONGRUENCES    WHOSE   MODULI   ARE   IDEALS. 

Hence 

iK'i)  +<H'2)  H hiKf»)  =*d)  +  <K'2)  H h  *('»).  14) 

Since,  however,  every  term  in  the  first  member  of  14)  is  equal 
either  to  the  corresponding  term  in  the  second  member  or  to  o, 
and  hence,  if  even  a  single  term  in  the  first  member  of  14)  were 
o,  14)  would  not  hold,  no  term  in  the  first  member  of  14)  is  o. 

Therefore  f(t)  =<f>(t). 

An  examination  of  the  table  (p.  393)  will  illustrate  this. 

§27.  Primitive  Numbers  with  respect  to  a  Prime  Ideal 
Modulus.1 

Among  the  integers  of  a  reduced  remainder  system,  mod  p, 
there  are,  we  have  seen,  <£(<£(£))  that  belong  to  the  exponent 
<f>(p).  These  integers  are  caller  primitive  numbers  with  respect 
to  the  modulus  p,  or  briefly,  primitive  numbers,  mod  p. 

From  the  table  (p.  393)  we  see  that  3  and  5  are  primitive  num- 
bers with  respect  to  the  modulus  (7,  3  +  V —  5)-  If  p  be  a  primi- 
tive number,  mod  p,  the  <f>(p)  powers  of  p, 

n°=  I    n1    n2    n3    •••    n^^)-1 
P    x>  P  }  P  >  R  >         >  P  y 

form  a  reduced  residue  system,  mod  p.  This  is  for  many  pur- 
poses an  extremely  useful  way  of  representing  such  a  system. 

We  can  determine  a  primitive  number,  mod  p,  by  the  method 
used  (Chap.  Ill,  §  33)  to  determine  a  primitive  root  of  a  rational 
prime. 

We  can  prove  Wilson's  Theorem  for  an  ideal  modulus  by  the 
aid  of  such  a  reduced  residue  system,  just  as  the  original  theorem 
was  proved  for  rational  integers  (Chap.  Ill,  §  29). 

It  will  be  noticed  that  the  primitive  numbers,  mod  p,  play  exactly  the  same 
role  with  regard  to  p  that  the  primitive  roots  of  a  rational  prime,  p,  do 
with  regard  to  p.  It  would  seem  desirable  to  have  the  nomenclatures  the 
same,  but  those  employed  are  the  usual  ones.  It  would,  perhaps,  be  best 
to  use  the  term  primitive  number  instead  of  primitive  root  in  the  case 
of  rational  integers. 

§  28.    Indices. 

//  CL  =  pl,  mod  p, 

'See  Hilbert:  Bericht,  §9. 


CONGRUENCES    WHOSE    MODULI    ARE   IDEALS. 


399 


where  p  is  a  primitive  number,  mod  p,  and  i  be  one  of  the  num- 
bers o,  I,  2,  .--,  <p(p)  — i,  i  is  said  to  be  the  index  of  a  to  the 
base  p  with  respect  to  the  modulus  p. 

The  relation  between  an  integer  and  its  index,  which  was  seen 
in  R  to  be  similar  to  that  of  a  number  to  its  logarithm,  is  evidently 
the  same  in  the  case  of  ideals.  It  can  be  shown  exactly  as  in 
R  that,  if  p  be  any  primitive  number,  mod  p,  a,  (3  any  integers  of 
the  realm,  and  m  a  positive  rational  integer,  we  have  the  follow- 
ing relations. 

i.  The  index  of  the  product  of  two  integers  is  congruent  to  the 
sum  of  the  indices  of  the  factors,  mod  <j>(p),  that  is; 

indp  (aft)  =  indp  a  +  indp  /?,  mod  <f>(p) . 

ii.  The  index  of  the  mth  power  of  an  integer  is  congruent  to  m 
times  the  index  of  the  integer,  mod  <f>(p),  that  is; 

indp  am  =  m  indp  a,  mod  cp(p). 

We  observe  that  in  every  system 

indp  I  =  o. 

By  means  of  the  following  tables  we  can  illustrate  the  use  of 
indices  for  an  ideal  modulus.  Table  A  gives  for  the  modulus 
(7>  3  +V — 5)  the  index  to  the  base  3  of  each  integer  of  a 
reduced  residue  system,  and  Table  B  gives  the  residue  corre- 
sponding to  any  index  to  the  same  base  and  modulus. 

It  is  evident  that  two  integers  congruent  to  each  other,  mod  p, 
have  the  same  index  in  any  system  of  indices,  mod  p. 

A. 


Residue 

1 

2 

3 

4 

5 

6 

Index 

0 

2 

1 

4 

5 

3 

B. 

Index 

0 

1 

2 

3 

4 

5 

Residue 

» 

3 

2 

6 

4 

5 

400  CONGRUENCES    WHOSE    MODULI    ARE   IDEALS. 

To  pass  from  an  index  system  with  the  base  px  to  one  with  the 
base  p2,  the  modulus  being  p,  we  find  as  in  R  that 

indp2  a  =  indpi  a  •  ind^  pv  mod  <p(p) ; 

that  is,  to  obtain  the  system  with  base  p2  from  one  with  base  plt 
we  multiply  each  index  of  the  latter  system  by  indp  plf  the  smallest 
positive  residues,  mod  <p(p),  of  these  products  bring  the  required 
system  to  the  base  p2. 

In  particular,  if  a  =  p2,  we  have 

ind^/yind^EEE  i,  mod  </>(». 

Ex.  To  obtain  for  the  modulus  (7,  3  +  V —  5)  a  system  of  indices  to 
the  base  5  from  one  of  the  base  3  we  have  first  to  find  ind53.  From  the 
relation  just  given 

ind3  5  •  ind.,  3^  1,  mod  6, 

whence  from  Table  A  it  follows  that 

5  ind5  3^1,  mod  6, 
and  therefore 

indi  3  =  5- 

Multiplying  by  5  each  index  to  the  base  2  and  taking  the  least  posi- 
tive residues,  mod  6,  of  these  products,  we  obtain  for  the  modulus 
(7,  3  +  V  —  5)  the  following  table  of  indices  to  the  base  5. 


Residue    |   1 

2 

3 

4 

5 

6 

Index          0 

4 

5 

2 

1 

3 

§  29.    Solution  of  Congruences  by  Means  of  Indices. 

As  in  R,  the  solution  of  any  congruence  of  the  form 

ax  =  /3,  mod  p,  1) 

where  a  is  not  divisible  by  p,  can  be  effected  by  means  of  a  table 
of  indices  for  the  modulus  p ;  for  from  1 )  it  follows  that 

ind a  +  hid •*"  =  ind (3,  mod  4>(p), 
which  gives 

ind  x=  ind  (3  —  ind  a,  mod  <f>(p), 
from  which  x  can  be  determined. 


CONGRUENCES    WHOSE    MODULI    ARE   IDEALS.  4OI 

Ex.  i.     From  the  congruence 

(2  +  V— ^arsss—  I+SV^,  mod  (7,  3  +  V^), 
we   obtain   ind3    (2  +  V  —  5) -}- ind3  ^^ind3    ( — I+3V  —  5),   mod  6; 
that  is,  since 

2  +  V"3r5  =  6,  mod  (7,  3  +  V=r5), 
and 

—  I^V^ss*  mod  (7,  3  +  V^r5), 

3  +  ind3  ;r  ^  4,  mod  6, 
or 

inda  X  =  I, 
whence 


*==3,  mod   (7,  3+ V- 5)- 
The  solution  of  the  congruence 

cur*  mm  f},  mod  p,  2) 

where  a  is  not  divisible  by  p,  can  be  reduced  by  the  use  of  indices 
to  the  solution  of  a  congruence  of  the  first  degree,  mod  <j>(p). 
From  2)  it  follows  that 

inda-j-  winder  =  ind/?,  mod  cf>(p), 
and  hence 

n'mdx  =  md/3 —  ind  a,  mod  <j>(p),  3) 

which  is  a  congruence  of  the  first  degree  in  the  unknown  x. 
Moreover,  n,  ind  x,  ind  /?,  ind  a  and  <f>(p)  are  evidently  to  be 
regarded  merely  as  integers  of  R.  Hence  by  §  14  the  necessary 
and  sufficient  condition  that  3)  shall  be  solvable,  is  that  ind  ft 
—  ind  a  shall  be  divisible  by  the  greatest  common  divisor,  d,  of 
n  and  <f>(p),  and,  if  this  condition  be  satisfied,  3)  has  \d\  roots. 

To  these  \d\  values  of  ind  x  correspond  \d\  values  of  x  satis- 
fying 2)  and  incongruent,  mod  p.  These  are  the  roots  of  2). 
We  see  therefore  that  by  the  use  of  a  table  of  indices  we  can 
reduce  the  solution  of  both  1)  and  2)  to  the  solution  of  con- 
gruences between  rational  integers. 

Ex.  2.     Consider  the  congruence 

(i  +  V^)*4^—  V^5,  mod  (7,  3  +  Vzr5),  4) 

where  1  +  V  —  5  is  not  divisible  by  (7,  3  +  V  —  5)  • 
ind3  (1  +  V  —  5)  +4  ind3  x^=  ind3  —  V  —  5,  mod  6;  that  is,  since 

i-hV^SssS.    mod    (7,    3  +  \A=r5), 
26 


4-02  CONGRUENCES    WHOSE    MODULI   ARE   IDEALS. 

and 

— V— $ss3.  mod    (7,  3  +  ^/^5), 
using  table  A, 

5  +  4  ind3  x  ^=  1,  mod  6 
or 

4  ind3  x^2,  mod  6.  5) 

Since  the  greatest  common  divisor,  2,  of  6  and  4  divides  2,  the  con- 
gruence 5)  has  two  roots  which  are  easily  found  to  be  2  and  5. 
Hence  we  have 

ind3  j  =  2  or  5, 
and  therefore 

x==2  or  5,  mod  (7,  3  +  V  —  5). 

These  results  are  easily  verified  by  substitution  in  4).    We  obtain 

(1  + V^)24  =  2  +  2V^r5==  —  V^5,  mod  (7,  3  +  V^5), 
and 

(i  +  V^5)5*  =  2  +  2VTir5  =  —  V^5,  mod   (7,  a  +  V^1!)- 

Ex.  3.    The  congruence 

(1  +  Vz=r5)^^2,  mod  (7,  3  +  V^r5), 

has  no  roots,  since  the  congruence 

ind3  (1  +  V  — 5)  +4  md3  *  =  ind3  2,  mod  6, 
or 

4  ind3  x  s  3,  mod  6, 

has  no  roots,  the  greatest  common  divisor,  2,  of  4  and  6  not  dividing  3. 
Ex.  4.     Construct  a  table  of  indices  to  the  base  10  for  the  modulus 
(23,  8  -f-  V  —  5)  and  solve  by  its  aid  the  congruence 

(2-\-3yy-~5)^  =  -^/^5,  mod   (23,  S  +  V"11!). 
Ex.  5.     Show  that  the  congruence 

(i  +  V=S)**e=i5,  mod  (23,  8  +  V^5) 
has  no  root. 

The  congruence  xn  s  j3,  mod  p,  where  p  is  a  prime  ideal,  can  be  treated 

as  was  the  corresponding  congruence  in  R  (Chap.  Ill,  §  34),  and  a  criterion 

for  its  solvability  given  analogous  to  Euler's.     The  general  congruence  of 

the  2d  degree  in  one  unknown  can  be  discussed  and  the  first  part  of  the 

theory  of  quadratic  residues  for  ideal  moduli  developed  as  in  R,  Legendre's 

symbol  being  replaced  by  (  -  J ,  where  a  is  an  integer  and  p  a  prime  ideal 

of  k(Vm)  (see  Sommer:  Vorlesungen  iiber  Zahlentheorie,  pp.  92-98). 
The  reader  should  work  out  the  above.  It  is  evident  from  the  nature 
of  an  ideal  that  no  direct  reciprocal  relation  can  exist  between  a  and  p, 
such  as  that  between  two  rational  primes  as  expressed  bv  the  quadratic 
reciprocity  law.  A  discussion  of  the  reciprocity  laws  in  the  higher  realms 
is  beyond  the  scope  of  this  book;  for  them  the  reader  may  consult  Hilbert : 
Bericht,  and  Math.  Ann.,  Vol.  51;  Sommer:  V.  u.  Z.,  Fiinfter  Abschnitt. 


CHAPTER   XIII. 
The  Units  of  the  General  Quadratic  Realm. 

§  i.    Definition. 

The  units  of  any  quadratic  realm  are  those  integers  of  the 
realm  which  divide  every  integer  of  the  realm.  For  purposes  of 
investigation  they  may  be  defined  as  follows : 

i.  The  divisors  of  I  and  hence  those  integers  whose  recip- 
rocals are  integers. 

ii.     Those  integers  whose  norms  are  ±  I. 

These  two  definitions  are  easily  seen  to  coincide ;  for,  if  c  be  a 
unit  of  &(Vw),  we  have  from  i 

ea=i,  i) 

where  a  is  any  integer  of  k(\/m). 
From  i)  it  follows  that 

«[e]«[a]  =  I, 

and  hence  n[e]  =  ±  I ; 

that  is,  ii  is  a  consequence  of  i. 

Likewise,  if  e  be  a  unit  of  &(\/m),  we  have  from  ii 

ee'=±I, 

where  e  is  the  conjugate  of  c  and  therefore  an  integer  of  k(^m). 
Therefore  e  is  a  divisor  of  I,  and  hence  i  is  a  consequence  of  ii. 
It  follows  from  the  above  definition  that  if  each  of  two  integers, 
a,  /?,  divide  the  other,  their  quotient  is  a  unit;  for,  if 

a//?  =  y, 

y  and  i/y  are  both  integers ;  hence  y  is  a  unit  by  i.  In  particular, 
the  quotient  of  two  units  is  a  unit.  In  investigating  the  units  of 
the  general  quadratic  realm,  we  shall  distinguish  two  cases  accord- 
ing as  the  realm  is  imaginary  or  real. 

403 


4O4  THE    UNITS    OF   THE    GENERAL    QUADRATIC    REALM. 

§  2.    Units  of  an  Imaginary  Quadratic  Realm. 

The  fact  that  the  norms  of  all  the  integers  of  an  imaginary 
quadratic  realm  are  positive  will  enable  us  to  determine  the  units 
of  such  a  realm. 

Let  m  be  a  positive  integer  containing  no  squared  factor;  then 
&(V — m)  is  an  imaginary  quadratic  realm,  and  we  have  seen  that 
all  imaginary  quadratic  realms  will  be  obtained  if  m  take  all 
positive  values. 

Let  e,  =  x -\-yio,  be  a  unit  of  k(\/ — m),  1,  w  being  a  basis  of 
the  realm. 

We  have 

n[€]  =  (x-\-yo>)(x  +  y»')  =  i,  1) 

the  value  —  1  being  impossible,  since  the  realm  is  imaginary. 

We  have  now  to  see  for  what  rational  integral  values  of  x 
and  ti)  holds,  and  to  do  so  must  distinguish  two  cases. 

i.     When  — m  =  2  or  3,  mod  4,  and  hence  w=y — m. 

Then 


n [e]  =  (x  -L-  y  V —  m)  (x  —  y  V —  m)  =  x2  -\-  my2  =  1. 

If  m  >  1,  it  follows  that  y  =  o  and x=±  1,  and  hence  c  =  ±  1. 
If  m=i,  we  have  the  realm  k(i)  whose  units  we  have  found 
to  be  ±  1,  ±  i. 


ii.     When  — fnasi,  mod  4,  and  hence  <a=  (1  +V — m)/2. 
Then 

1  -f  V  —  m\f  1  —  y/—m\ 


r  .,       /  1  +  V  —m\( 


2     my1 
+  —  -*  I- 


4 
If  m  >  4,  it  follows  that  y  =  o  and  a-=±  i,  and  hence 

c=±  1. 

If  w  =  3  we  have  the  realm  fc(V — 3)  whose  units  we  have 
found  to  be  ±  1,  =b[(i  ±V — 3)/2]-  We  see,  therefore,  that 
k(i)  has  the  four  units  ±  /,  ±i,  and  k(y — 3)  the  six  units 
±  1,  db  [(1  ±V — 3)/2]>  and  that  all  other  imaginary  quadratic 
realms  have  only  the  two  units  ±  /. 


THE    UNITS    OF    THE    GENERAL    QUADRATIC    REALM. 


405 


§  3.     Units  of  a  Real  Quadratic  Realm. 

The  determination  of  the  units  of  a  real  quadratic  realm  is 
much  more  difficult.  We  shall  see  that,  as  in  the  realm  k(^/2), 
the  units  of  such  a  realm  are  infinite  in  number  and  can  all  be 
expressed  as  powers  of  a  single  unit  called  the  fundamental  unit. 
To  show  this  we  shall  need  the  two  following  theorems,  the  first 
of  which,  due  to  Minkowski,  is  of  great  importance  in  the  theory 
of  numbers. 

Theorem  i.  //  clxx -\- pxy,  a2x-\-/32y  be  two  homogeneous 
linear  forms  with  real  coefficients  whose  determinant 

CL      ft 


8  = 


is  not  0,  there  exist  tzvo  rational  integers,  x0,  y0,  not  both  zero 
such  that 


and 

If  we  put 

then 


|<*i*o  +  A:yo|i|V8|, 
1*2*0 +  &yolslV*|.1 


x=^t 


A.     A 


1) 


7, 


or 


Putting 


y^A.i  +  B.r, 


B. 


2) 


we  see  that  A8=  1. 

If  now  we  can  find  two  quantities,  £0,  r]0,  such  that 

I&J^I/IVA]  and   ho|^i/|VA[, 
1  Minkowski:  Geometrie  der  Zahlen,  p.  104.     Hilbert:  Bericht,  Hulfsatz  0. 


406  THE    UNITS    OF   THE    GENERAL   QUADRATIC    REALM. 

and  such  that  the  corresponding  values  x0,  y0  of  x  and  y  are 
rational  integers,  then  x0  and  y0  are  the  required  values  of  x  and  y. 


For,  if 

^'o^^i^o  +  ^o. 

and 

3'0  =  ^2&  +  -^2'70» 

then 

rj0  =  a2x0  +  p2y0, 

and  hence, 

since 

|«o|^|V«l  and  h0|= 

we  have 

l^-o  +  tool^lVSl. 

To  prove  our  theorem  it  will  be  sufficient  therefore  to  show 
that  two  quantities,  |0,  r}0,  exist  which  satisfy  the  conditions 

|€o|  ^i/IVA| ;    hfo|si/|VS|, 

and  such  that 

are  rational  integers,  where  Alt  A2,  Bx,  B2  are  real  and 


A  = 


A     B, 

A   b2 


4=0. 


In  the  proof  of  the  theorem  we  shall  prove  first  the  case  in 
which  alt  a2,  filf  (32  are  rational  and  integral,  then  that  in  which 
the  coefficients  are  rational  and  finally  require  merely  that  they 
be  real.  In  the  first  two  cases  the  theorem  will  be  proved  in  its 
original  form,  in  the  last  case  in  the  equivalent  form  given  above. 

The  proof  in  the  second  case  will  depend  directly  upon  the 
truth  of  the  theorem  for  the  first  case,  and  that  in  the  third  case 
upon  case  two. 

i.     Let  alf  a2,  f}lf  fi2  be  rational  integers. 

We  shall  need  a  theorem  concerning  binary  linear  forms. 

Calling  a  binary  linear  form  axx  +  bxy,  where  alf  bx  are  ra- 


THE   UNITS    OF   THE    GENERAL    QUADRATIC   REALM.  407 

tional  integers,  for  the  sake  of  brevity  a  form,  and  two  such  forms 
a  form  system,  we  say  that  a  form  cxx  -f-  dxy  is  reducible  to  o  by 
the  form  system  axx  +  bxy,  a2x  -f-  b2y,  if 

cxx  +  dxy  =  gx(axx  +  bxy)+g2(a2x  +  b2y), 

where  gx,  g2  are  rational  integers. 

Two  forms  are  reducible  to  one  another  by  a  given  form  system 
if  their  difference  is  reducible  to  o  by  this  system. 

Two  form  systems  are  said  to  be  equivalent  if  every  form  that 
is  reducible  to  o  by  either  one  of  the  systems  is  also  reducible  to 
o  by  the  other  system. 

The  analogy  to  the  basis  of  an  ideal  is  at  once  evident,  for,  if 
ax  s=s  (axu>x  +  bxw2,  a2(ox  -f-  b2oy2)  be  an  ideal,  where  ax<ax  -j-  b2o)2, 
a2o)x  -f-  b2o)2  is  a  basis,  then  an  integer,  cx(ox  -)-  dxa>2,  is  a  number  of 
the  ideal  if 

cx(ox  +  dx<o2  =  gx(ax(ox  +  bx(o2)  +  <72(a2wi  +  o2<o2), 

where  gx,  g2  are  rational  integers.  Thus  the  reducibility  of  a 
form  to  o  by  a  given  form  system  corresponds  to  a  number  be- 
longing to  an  ideal. 

We  can  show  exactly  as  in  the  case  of  a  canonical  basis  of  an 
ideal  (Chap.  XI,  Th.  i)  that  among  the  form  systems  equivalent 
to  a  given  system  there  is  one,  Ax,  Bx  -f-  Cy,  such  that  among 
all  forms  of  the  form  ax,  reducible  to  o  by  the  given  system,  Ax 
is  that  one  in  which  a  is  smallest  in  absolute  value,  and  among 
those  of  the  form  bx  -f-  cy  reducible  to  o  by  the  given  system, 
Bx  -\-  Cy,  is  one  of  these  in  which  c  is  smallest  in  absolute  value. 
We  can  then  show  that,  if  two  form  systems  be  equivalent,  the 
absolute  values  of  the  determinants  of  their  coefficients  are  equal 
(see  Chap.  XI,  Th.  i,  Cor.). 

It  will  now  be  evident  that  to  say  in  the  case  of  forms  that 
two  forms  are  reducible  to  one  another  by  a  given  form  system 
is  the  same  as  saying  in  the  case  of  an  ideal  that  two  integers  are 
congruent  with  respect  to  this  ideal,  for  in  the  former  case  the 
difference  of  the  two  forms  is  reducible  to  o  by  the  given  system 
while  in  the  latter  the  difference  of  the  two  integers  is  a  number 
of  the  ideal. 


4o8 


THE    UNITS    OF   THE    GENERAL    QUADRATIC    REALM. 


The  statement  in  the  one  case  that  there  are  exactly 


ai    K 

a2    K 


forms,  no  two  of  which  are  reducible  to  one  another  by  the  form 
system  axx  +  bxy,  a2x  -\-  b2y,  is  the  same  as  the  statement  in  the 
other  case  that  there  are  exactly 

a2        b2 

integers  which  are  incongruent  each  to  each  with  respect  to  the 
ideal  {axux  +  a2w2,  bxwx  -f-  b2o>2),  and  may  be  proved  similarly  (see 
Chap.  XII,  Th.  i). 

We  observe  now  that  |8|  is  equal  to  one  of  the  square  numbers 

i,  4,  9,  1 6,  25,  -..,  r2,  (r+1)2, 

or  lies  between  two  of  them. 

Let  r*g|*f<  ('+l)a. 

and  form  the  (r+  i)2  forms 


ra  =  o,i,2,...,r, 
ax  +  by  A 

1     J   \  b  =  o,  1,2,  ->,r. 


3) 


Since  there  are  only  \B\  forms,  no  two  of  which  are  reducible 
to  one  another  by  means  of  the  form  system  axx  -\-  (3xy,  a2x  -f-  (32y, 
at  least  two  of  the  forms  3)  are  reducible  to  one  another  by  this 
system. 

Let  these  two  forms  be  a\x  -\-  biy  and  tyx  -\-  bjy. 

Then 

dix  +  bty  =  ajx  +  b$y  +  c(axx  +  (3xy)  +  d(a2x  +  fty ) ; 
that  is, 

(at  —  a,)*  +  (bi  —  bj)y=  (axc  +  a2d)x  +  (ftc  +  ftd)y, 
and  hence  o^c  +  a2d  =  a<  —  a,-, 


THE    UNITS    OF   THE    GENERAL    QUADRATIC    REALM. 


409 


Since  \ai  —  aj\  and  |&«  —  bj\  -^r,  they  are  both  g  | \/8\ ;  hence  c 
and  d  are  the  required  values  of  x  and  y. 

ii.     Let  at,  a2,  ft,  ft  be  rational  fractions. 

Let  their  least  common  denominator  be  g.  Then  #ax,  #a2,  #ft, 
</ft  are  rational  integers. 

By  case  i  we  can  find  two  rational  integers,  x0,  y0,  such  that 

\g<*z*o  +  #ft3'o|^  I  V¥2|-  4) 

On  dividing  both  members  of  4)  by  g  we  get 

k*0  +  ft3'o|^|V8|, 

K*o  +  ft3'o|^|V8|. 

Hence  x0  and  y0  are  the  required  values  of  x  and  y. 
iii.     Let  alt  a2,  ft,  ft  be  any  real  numbers. 
We  shall  prove  the  theorem  in  its  second  form ;  that  is,  that  if 
Alf  A2,  Blt  B2  be  any  real  numbers,  such  that  the  determinant, 

A   b2\ 

is  not  zero,  there  exist  two  numbers,  £0,  rj0,  satisfying  the  conditions 

|&|*X/|V*|.      ho|^l/|VA|, 

and  such  that  x0  =  A1£0  -f-  Bxr)0, 

y0=A2$0-\-B2rj0, 
are  rational  integers. 

Let  Alf  A2,  Blf  B2  be  defined  respectively  by  the  rational  fun- 
damental series 

alt  a2,  az,  •• 

blt  b2,  b3,  ■■ 
Ox's  <*2>  a%>  ■  • 


5) 


that  is, 


^1  =  lim  an,     B1=lhn  b, 


At>  =  \im  (in,     J5„  =  lim  bj 


6) 


410  THE    UNITS    OF   THE    GENERAL    QUADRATIC    REALM. 

Let 

A  = 


a       b' 


where  an,  bn,  an',  bn'  are  the  nth  terms  of  the  above  series, 
then 

lim  An  =  lim  an  •  lim  bn'  —  Hm  anr  •  Hm  bn, 

=  A1B2  —  A2B1  =  &. 

We  observe  now  that  in  the  series 

*  Ax,  A2,  A3,   -..,  7) 

though  some  of  the  terms  may  be  o,  the  number  of  such  terms  is 
always  finite ;  that  is,  from  some  ith  term  onward  no  A  is  o ;  for 
otherwise,  lim  An  would  not  exist  or  else  would  be  o. 

Since  now  the  terms  5)  are  all  rational  numbers,  and  A*  and 
all  succeeding  A's  are  different  from  o,  we  can  find  by  case  ii 
for  every  set,  ai+p,  bi+p,  a'i+p,  b'i+p  of  (i-{-p)th  terms  of  the  series 
5),  two  numbers,  &+p,  rji+p,  such  that 

|&*| s-i/! V^Ji#|..    h*+p|ii/|V^|,         ,      8) 

and  that  0* +p&+p  +  bi+v-qi+p, 

a'i+p£i+P  +  bfi+PY)i+Pf 

are  rational  numbers. 

From  8)  it  is  evident  that  the  terms  of  the  series 

rji,  rji+1,  rji+2,    •••, 

have  an  upper  limit,  for  no  term  of  the  series 

|Ai|,   |A{+1|,    |A4+2|,   -.., 
is  o,  and  lim  An  =  A  =4=  o,  whence  the  terms  of  this  series  have  a 

lower  limit. 

Let  this  upper  limit  of  the  £'s  and  */s  be  k. 

Consider  a  system  of  rectangular  axes  and  construct  a  square 


THE    UNITS    OF   THE    GENERAL    QUADRATIC    REALM, 


41 


with  the  origin  as  center,  its  sides  equal  to  2k  and  parallel  to  the 
axes. 

v 


If  now  we  consider  £i+p,  rji+p  as  the  abscissa  and  ordinate  re- 
spectively of  a  point,  we  may  represent  each  pair  of  numbers 
£up,  yi+p(P  =  o,  1,  2,  •  •  • )  by  a  point. 

All  these  points  will  be  within  or  on  the  boundary  drawn  as 
above. 

Since  there  are  infinitely  many  points  (&+p,  r}%+v)  within  or  on 
this  boundary  they  will  have  at  least  one  limiting  point  within 
or  on  the  boundary.  Let  the  coordinates  of  this  point  (or,  if 
there  be  more  than  one,  of  any  particular  one)  be  £0,  rj0. 

There  will  be  certain  series  of  the  points  (|i+p,  r)i+p)  which 
approach  and  remain  arbitrarily  close  to  (£0,  rj0)  as  p  is  indefi- 
nitely increased. 

If  (f^y,  yi+  /)  denote  such  a  series,  where  p'  represents  only 
those  values  of  p  which  gives  this  series,  we  have 

£0  =  lim  £*+,'»  %  =  lim  W 


Then 


P'=o 


lim  ("i+Jiw  +  **f«*w)  =  AS0  +  B%> 

p'=x> 

•im  «,yf„y  +  b\+p,r,.+p)  =  A%  +  B'%. 

P'=«3 

But  all  terms  of  the  series 


and 


are  rational  integers. 


a'i+A+p'  +  ^i+pViW 


412  THE    UNITS    OF   THE    GENERAL    QUADRATIC    REALM. 

Hence  their  limits,  A$0  -f-  B-qQ  and  A'£0  -f-  B'-q0i  are  rational  in- 
tegers. Therefore  £0  and  r/0  are  the  required  numbers,  and  the 
theorem  is  proved  in  its  second  form.  It  holds  therefore  in  its 
original  form. 

From  the  above  theorem  we  have  at  once  the  following  theorem : 

Theorem  2.  If  axx  +  fSxy,  a2x  -f-  (32y  be  two  homogeneous 
linear  forms  with  real  coefficients,  whose  determinant 

«i     Pi 

"1     ft 

is  not  0  and  k,  kk  be  any  two  positive  quantities  such  that 

K*k=  |8I> 
there  exist  rational  integers  x0,  y0,  not  both  0,  such  that 

Given  the  two  forms 


Lft  a2       ,    ft 


1. 


whose  determinant  is  not  zero,  there  exist  by  Th.  1  two  rational 
integers,  -x0,  yoy  not  both  o,  such  that 

|*i    ,  ft   u 

\fC        °  K  J°\  ' 


«.  _L    ft 

Z^  +  IT* 


^  1 


'A  ,VA 

and  hence  l^i^'o  +  Ay0|  g  *, 

|*t*t + toils  «* 

From  this  theorem  we  obtain  at  once  the  following  theorem, 
which  is  necessary  for  the  investigation  concerning  the  units  of 
a  real  quadratic  realm  as  well  as  interesting  on  its  own  account. 

1  Hilbert :  Bericht,  Hulf  satz  7. 


=  a/  —  tu  =  Vd  4=  O, 


THE    UNITS    OF   THE    GENERAL    QUADRATIC    REALM.  413 

Theorem  3.  There  are  in  every  real  quadratic  realm  an  in- 
finite number  of  integers,  the  absolute  value   of  whose  norms 

i.  The  existence  of  at  least  one  such  integer  is  seen  at  once. 
For,  if  1,  to  be  a  basis  of  the  realm, 

x  -f-  yoj,     x  +  y*' 

are  two  linear  forms  whose  determinant 

1      &) 

1     *»'! 
and  making  use  of  Th.  2  and  putting 

K=KX,         K^   =   \\/'d\/K1, 

where  *t  >  o,  we  see  that  there  exist  two  rational  integers,  xx,  ylf 
which  are  not  both  o,  and  which  are  such  that 

k  +  ^'I^IWlAi, 

and  hence 

I  {*i  +  &•)  Oi  +  3'X)  f  s  I  V^| ; 

that  is, 

M*i+*rf|*'|V2|r 

Therefore  the  realm  contains  at  least  one  integer,  a1,  =  .r1  +  >'iw> 
the  absolute  value  of  whose  norm  is  less  than  or  equal  to  |V^|- 

To  show  that  there  are  an  infinite  number  of  such  integers  we 
proceed  as  follows : 

To  prove  the  existence  in  the  realm  of  an  integer,  a2,  =  x2-\-  y2o>, 
that  is  different  from  ±  alt  and  such  that 

\n[a2]\^\Vd\, 
we  have  only  so  to  choose  k,  that  from  the  condition 

it  will  necessarily  follow  that 

a2^±av 


4 14  THE   UNITS   OF   THE   GENERAL   QUADRATIC   REALM. 

This  may  be  effected  in  infinitely  many  ways,  a  simple  one  being 
to  take   for  k±   some  positive  quantity  /c2<|a1|;    for   example, 

\aj2\,  for  then  from  \^2[^K2<Kx 

it  follows  that  |a,|  <  \ax\, 

whence  a2  =j=  ±  ol±. 

Since  by  Th.  2  there  exist  two  rational  integers,  x2,  y2,  which 
are  not  both  o  and  which  are  such  that 

\x2  +  y2«>  |^K2, 

it  follows  that  there  is  in  the  realm  an  integer,  a2,  =  x2  -f-  y2w, 
different  from  dfc  aly  and  such  that 

|»la,]|s|V3|. 

To  prove  the  existence  in  the  realm  of  a  third  integer,  a3,  dif- 
ferent from  ±  a.!  and  ±  a2  and  such  that 

we  have  only  to  put  for  k  in  the  inequality  a  positive  quantity  k3 
less  than  k2,  when  it  is  at  once  evident  that  such  an  integer 

exists;  for  from  \a3\  g*3  <[a2|<Iail 

it  follows  that         a3  =f=  ±  a2,  and  a3  4=  ±  ax. 

Continuing  in  this  manner  we  can  prove  the  existence  in  the 
realm  of  as  many  such  integers  as  we  choose.  They  are,  there- 
fore, infinite  in  number. 

Ex.  We  shall  illustrate  the  above  theorem  by  showing  that  we  can 
actually  find  in  k{yjy)  as  many  integers  as  we  please,  the  absolute  values 
of  whose  norms  are  less  than  or  equal  to  |  V2^  I,  d  being  in  &(V7) 
equal  to  28. 

Following  the  method  employed  in  the  proof,  we  let  a.i,  =  Xi  -\-  yiV7> 
be  any  integer  satisfying  the  required  condition  and  *  be  any  positive 
quantity,  say  2. 

We  have  to  determine  x,  y  so  that 

\a,\i^Bi,     W\f  9) 


THE    UNITS    OF   THE    GENERAL    QUADRATIC   REALM.  415 

We  may  assume  without  loss  of  generality  that  Xi,  yi  have  the  same 
sign,  for,  this  assumption  being  made  and  V7  being  taken  positive, 

I  X!  +  yiV7 1  >  I  xi  —  W7  I, 

otherwise  not,  and  the  most  favorable  way  in  which  the  conditions  9) 
can  be  imposed  is  |  xx  +  .ViV7  I  = tne  larger  of  the  two  quantities  k  and 
I  V  d  I  A,  here  |  V2&  |/2,  |  Xi  —  yi\/7  |  ^the  smaller  of  the  two  quantities 
k  and  I  V^  |/K>  here  2.  Making  this  assumption,  the  conditions  which 
Xi,  yi  must  satisfy  are 

|*t-f  W7|iV7,  10) 

\x1  +  y1y/7\^2.  11) 

The  further  assumption  Xi,  yi  positive,  which  may  evidently  be  made 
without  loss  of  generality,  will  simplify  the  work. 
Doing  this,  we  see  that,  since  Xi  and  yi  have  the  same  sign, 


*i  =  0, 

yi  =  i 

Xi=z  I, 

yi  =  o 

Xi  =  2, 

yi  =  o. 

or 

or 

But  it  is  evident  from  11)  that  of  the  three  values  only  those  pairs  in 
which  yi  =  0  are  admissible ;  hence 

xi  +  yiV7  =  I  or  2. 

The  only  integers  of  k{\/j)  which  satisfy  the  condition  9)  are  therefore 
±  1,  ±2.  The  absolute  values  of  the  norms  of  1,  — 1,  2,  and  — 2  are 
evidently  all  less  than  |  V28  |. 

To  find  another  integer  a«,  =  x2  +  y2\/7,  the  absolute  value  of  whose 
norm  is  less  than  |  V28  |»  we  proceed  as  in  the  proof  of  Th.  3  and  let 
k=  I  a,/2  I,  where  ax  is  any  one  of  the  integers  1,  — 1,  2  or  — 2,  say  2; 
that  is,  we  have  now  to  determine  x2,  y%  so  that 

I  x2  +  y2 V7 1  ^  V28.  12) 

1*2— y2V7l^i,  13) 

where  Xz,  y2  are  assumed  to  be  both  positive.  Since  x2,  y2  have  the  same 
sign  and  the  value  o  for  a2  is  excluded,  we  see  from  12)  that 

x2  =  0,  y2  =  1  or  2, 

or  #2=1  or  2,  y2  =  1, 

or  x2  s=  1,  2,  3,  4,  5,    y2  =  0 ; 

but  13)  excludes  all  these  values  except 

x2  =  2,        y2  —  1 
and 

x2  =  1,        y2  =  o. 

The  last  set  gives  a2  =  1,  an  integer  already  found,  but  the  other  gives 
cc2  =  2-j-  yjy,  a  new  integer  satisfying  the  conditions  12)  and  13),  and 
hence  one  the  absolute  value  of  whose  norm  is  less  than  |  V2^  |. 


4l6  THE    UNITS    OF   THE    GENERAL    QUADRATIC    REALM. 

We  see  indeed  that 

l»[2  +  V7]|=3<l  V&l 

If  now  we  put  *=s|  (2  —  V7)/2  l>  and  proceed  as  before,  we  can  find 
an  integer  a3  such  4hat 

I  n  [a3]  I  <  I  V28  I,  and  a,  =|=  ±  a*  a,  =)=  ±  a* 

Continuing  in  this  manner,  we  can  find  as  many  integers  as  we  please 
satisfying  the  required  conditions. 

Theorem  4.  //  *  be  any  positive  constant,  there  exist  only  a 
finite  number  of  algebraic  integers  of  the  second  degree  such  that 
they  and  their  conjugates  are  simultaneously  less  than  k  in  abso- 
lute value.1 

Let  a  be  an  integer  of  the  second  degree  such  that 

|a|  <*,     |.a'|  <K.  14) 

Let  x2  -\-  axx  -j-  a2  =  o 

be  the  irreducible  rational  equation  of  which  a  and  a'  are  the 
roots.     We  have 

a1  =  -*-(a-\-a'),    a2  =  aa', 

hence  |ax|=|a  +  a'|,     |a2|=|aa'|- 

But  \a-\-a'\<2K,     \aa'\<K2, 

hence  \ax\<2K,     |o2|<k2.  15) 

It  is  evident  that  only  a  finite  number  of  rational  integers  can 
satisfy  the  condition  15)  ;  hence  there  are  only  a  finite  number  of 
equations  of  the  second  degree  whose  roots  satisfy  14).  There 
are,  therefore,  only  a  finite  number  of  integers  of  the  second 
degree  satisfying  14).  This  theorem,  it  will  be  observed,  is 
proved  not  for  a  single  quadratic  realm  but  for  the  integers  of 
all  quadratic  realms  taken  together. 

Moreover,  it  will  be  noticed  that  not  all  the  roots  of  these  equa- 
tions satisfy  14)  but  that  among  their  roots  are  all  the  integers 
of  the  second  degree  that  satisfy  14).     See  Ex.  §  4. 

1  Hilbert :   Bericht,  Satz  43. 


THE    UNITS    OF   THE   GENERAL   QUADRATIC   REALM.  417 

Theorem  5.  There  exists  in  every  real  quadratic  realm  a 
unit,  e,  different  from  ±  1,  and  such  that  every  unit,  77,  of  the 
realm  has  the  form 

rj=±em, 

where  m  is  a  positive  or  negative  rational  integer,  or  o.1 

The  proof  of  this  theorem  may  be  conveniently  divided  into 
the  following  four  parts : 

i.  Every  real  quadratic  realm  contains  an  infinite  number  of 
integers,  alf  a2,  as,  •••,  the  absolute  values  of  whose  norms  are 
less  than  or  equal  to  \^Jd\. 

ii.  A  quadratic  realm,  whether  real  or  imaginary,  contains  only 
a  finite  number  of  ideals  whose  norms  are  less  than  \Vd\,  and 
hence  the  infinite  series  of  integers,  au  a2,  a3,  ••-,  considered  as 
principal  ideals,  (ax),  (a2)>  (as)>  '"»  9^ve  onh  a  finite  number 
of  different  principal  ideals,  whence  it  follows  that  the  integers, 
alf  a2,  a3,  •••,  must  fall  into  a  finite  number  of  classes,  each  con- 
taining an  infinite  number  of  integers  which  differ  from  each 
other  only  by  unit  factors,  and  hence  there  are  in  every  real  quad- 
ratic realm  an  infinite  number  of  units  different  from  ±  1. 

iii.  Infinitely  many  of  these  units  of  a  real  quadratic  realm  are 
greater  than  1 ;  among  these  there  is  a  smallest  one,  c. 

iv.  Every  unit,  rj,  of  the  realm  has  the  form 


where  m  is  a  positive  or  negative  integer,  or  0. 

Having  already  proved  i,  we  begin  with  ii. 

ii.  We  obtain  all  prime  ideals  whose  norms  are  less  than  |  \/d\ 
l)y  resolving  all  positive  rational  primes  less  than  | \/d\  into  their 
prime  ideal  factors. 

There  are  evidently  only  a  finite  number  of  such  prime  ideals. 
By  multiplying  these  prime  ideals  together  we  obtain  all  ideals 
whose  norms  are  less  than  |V^|-  These  ideals  are  evidently  also 
finite  in  number.  Hence  among  the  infinite  system  of  principal 
ideals 

(<*i)»(a2)»(a«)»  •'•>  16) 

aSee  Hilbert:  Bericht,  Satz  47. 
27 


41 8  THE    UNITS    OF   THE    GENERAL   QUADRATIC   REALM. 

whose  norms  g|  V^|>  at  least  one  ideal  must  be  repeated  an  infinite 
number  of  times. 

Let  the  infinitely  many  ideals 

(3*1)1  (a*2)>  (g#8)* 

taken  from  the  system  16)  be  the  same.     Then  each  one  of  the 
integers 

aiv  CLi2,  at3,  •••,  17) 

must  be  divisible  by  every  other  one ;  that  is,  we  have 

ail  =  /3ai2, 

and  ya.i1  =  a.i2, 

where  (3  and  y  are  integers. 

Hence  /3  and  y  are  units  (§  1),  and  are,  moreover,  different 
from  ±  1,  since  we  may  assume  that  no  two  of  the  integers  17), 
as  a*!,  cii2,  are  so  related  that 

3^=4=  ±  CLi2. 

Furthermore,  the  number  of  such  units  is  infinite ;  for 

ai1==Sai3f 

where  8  is  a  unit,  and  if  ft  =±8,  then  ai2=^±ai.i,  which  is 
impossible. 

Hence  the  quotients  obtained  by  dividing  each  of  the  integers 
17)  by  a«x  constitute  an  infinite  system  of  units, 

Vi,  %j  ■••> 

such  that  we  never  have 

r)i  =  ±  rfj. 

iii.  There  are  in  the  realm  an  infinite  number  of  units  which 
are  >  1 ;  for  from  each  one  of  the  units,  rflt  rj2,  •  •  •,  as  rji,  we  can 
derive  such  a  unit,  since  one  of  the  integers, 

all  of  which  are  units,  must  be  such  a  unit.     Among  this  infinite 
system  of  units  greater  than  1  there  is  a  smallest  one;  for,  if  rji 


THE    UNITS   OF   THE   GENERAL   QUADRATIC   REALM.  419 

be  a  unit  greater  than  i,  there  are  by  Th.  4  only  a  finite  number 
of  integers,  a,  of  the  realm  such  that 

|a|<i^;     |a'|<T7i; 

and  hence  only  a  finite  number  of  units,  rj,  such  that 

\y\<ii>    W\<vi-  l8) 

But  if  7]  be  any  unit  greater  than  1  but  less  than  rji,  we  have  from 

W=±  1, 

r?b'|=  I,  • 

and  hence  |?/|<i<  77*; 

that  is,  r]  must  satisfy  18). 

There  are,  therefore,  only  a  finite  number  of  units,  rj,  such  that 

and  hence  there  is  among  them  a  smallest  one,  which  is,  of  course, 
the  smallest  of  all  those  units  of  the  realm  that  are  greater  than  1. 
Denote  this  unit  by  e. 

iv.  It  is  evident  that  the  units 

•••,  ±e"2,  ±e-\  ±€°,  zte1,  ±€2,  -..,  19) 

are  all  different;  for  from 

em=±en,     m  >  n} 
it  would  follow  that 

em-n==z±.  If 

which  is  impossible,  since  e=4=±  1,  and  none  of  the  numbers  of 
the  realm  are  imaginary. 

We  shall  now  show  that  the  system  19)  comprises  all  units  of 
the  realm. 

Let  £  be  any  positive  unit  greater  than  or  less  than  1 ;  then  £ 
will  lie  between  two  consecutive,  positive  or  negative  powers  of 
c,  or  else  be  equal  to  a  power  of  e;  that  is,  we  can  determine  an 
integer,  n,  positive,  or  negative,  such  that 


€M<£<e 


n+l 


420  THE    UNITS    OF   THE    GENERAL    QUADRATIC    REALM. 

Let  !/€»=&; 

then  |i  is  a  unit,  and  we  have 

i  si  li  <  «• 

But  we  cannot  have 

K  &   <  €, 

for  6  is  the  smallest  unit  greater  than  i.     Hence 

and  therefore  £  =  en. 

When  n  is  positive  the  units  are  greater  than  i,  and  when  n  is 
negative  they  are  all  positive  but  less  than  i;  n  =  o  gives  £=ir 
By  letting  n  take  all  rational  integers  from  —  cc  to  -{-  cc  we  thus 
obtain  all  positive  units  of  the  realm. 

Now  let  £  be  a  negative  unit;  then  — £  is  a  positive  unit,  and 
we  have 

—  $=en; 

hence  $  =  —  en. 

Every  unit,  $,  therefore,  of  a  real  quadratic  realm  has  the  form 

where  n  is  a  positive  or  negative  rational  integer,  or  o,  and  c  is  the 
smallest  unit  of  the  realm  >  1. 

This  unit  e  is  called  the  fundamental  unit  of  the  realm. 

§  4.    Determination  of  the  Fundamental  Unit. 

If  in  any  quadratic  realm  k(\/m)  any  unit,  rj,  be  known,  we 
can  at  once  obtain  a  unit  greater  than  1 ;  for  one  of  the  four  units, 

rj,  —7),  i/rj  or  —  i/v, 
has  this  property. 

Denote  that  one  of  these  four  units  which  is  greater  than  1  by 
7)1.  We  have  now  to  determine  whether  there  are  any  units  in  the 
realm  which  are  greater  than  1  but  less  than  rju  and,  if  there  be 
any  such  units,  to  find  the  smallest  of  them. 


THE    UNITS   OF   THE    GENERAL    QUADRATIC   REALM.  42 1 

Th.  4  enables  us  to  do  this;  for  by  the  method  employed  in 
the  proof  we  can  find  the  rational  integral  equations  finite  in 
number,  among  whose  roots  are  the  integers  a  of  the  second 
degree  finite  in  number,  such  that 

\a\<*;     \a'\<m>  i) 

Among  these  integers  will  be  included  all  units,  |,  such  that 

K  !  <  %,  2) 

for  we  have  seen  that  from  2)  and 

££'=±h 
it  follows  that 

|f|<*;   |TK*. 

Since  we  wish  to  find  only  those  units  which  satisfy  1),  and 
the  last  term  of  the  irreducible  rational  equation  satisfied  by  an 
integer  of  the  second  degree  is  the  norm  of  the  integer,  we  may 
make  the  last  term  of  each  of  our  equations  =t  I. 

Writing  down,  therefore,  all  irreducible  equations  of  the  form 

x2  +  ax±  1=0,  3) 

where  a  is  a  rational  integer,  such  that 

\a\<2Vl, 

and  solving  these  equations,  we  obtain  all  units  which  satisfy  1), 
not  only  of  the  realm  under  discussion  but  of  all  real  quadratic 
realms. 

If  there  be  any  unit  of  the  realm  under  discussion  which  is 
greater  than  1  but  less  than  rj1}  it  will  be  found  among  these. 

Ex.    Let  the  realm  under  discussion  be  £(V5)-     Since 
«[2  +  V5~]=  —  1 

2  +  V5^is  a  unit  of  £(V5)-     Moreover  2-}-\/5>i- 

To  determine  those  units  of  &(V5)  that  are  greater  than  1  but  less 
than  2  -f-  y/s,  if  any  exist,  we  write  down  all  irreducible  equations  of  the 
form  3),  in  which  |  a  |  <  2(2 -|- V5)-  We  need  only  write  those  in 
which  a  is  negative  since  the  change  of  sign  of  a  merely  changes  the 
signs  of  the  roots. 


422  THE   UNITS   OF   THE   GENERAL   QUADRATIC   REALM. 

We   have,  therefore,   as   the   equations   among  whose   roots   will  be 
found  the  unit  sought  for,  if  it  exist, 


x-  —  X  +1  =  0 

x2—  x  —  J 

[  =0 

X2  —  2X  +  I  =  0 

X2  —  2X  —  ] 

[  =0 

x2  —  3x  +  I  =  o 

x2  —  2,x—] 

[  =0 

x2  —  4x  +  i  =  o 

x2  —  4x  —  ] 

[  =0 

x2  —  5*  +  i  =  o 

x2  —  $x  —  3 

[  =0 

x2  —  6x  +  i  =  o 

x*  —  6x—  J 

[  =0 

x2  —  7x  +  i  =  o 

x2  —  yx  — 

[  =0 

x2  —  8x  +  i  =  o 

x2  —  Sx  — 

[  =  0 

Solving  these  equations,  we  obtain  four  units  of  &(V5)  which  are  greater 
than  i,  £(i  +  V5XK3  +  V5),  2  +  V5  and  K7  +  3V5),  and  of  them 
evidently  £(i  +  V5)  1S  tne  smallest  and  hence  the  fundamental  unit. 

The  foregoing  determination  of  the  fundamental  unit  of  a  real 
quadratic  realm  depended  upon  the  supposition  that  some  unit 
of  the  realm  was  known.  To  find  some  unit  of  the  realm  we  may 
proceed  as  follows,  the  method  being  that  used  in  Th.  5  to  show 
the  existence  in  such  a  realm  of  a  unit  different  from  ±  1. 

Let  k(\Ztn)  be  the  realm.    0 

Determine  first  how  many  different  ideals  have  their  norms 
less  than  |\/d|.  This  is  easily  done  by  factoring  all  rational 
primes  less  than  |V^|  and  forming  all  products  of  these  ideals, 
such  that  the  norms  of  these  products  are  less  than  |yd|.  Sup- 
pose that  there  are  m  different  ideals  whose  norms  are  less 
than  |yd|. 

Find  now  w+i  integers  whose  norms  are  less  than  |Vd|, 
which  can  be  done  by  the  method  used  in  the  proof  of  Th.  3. 
The  quotient  of  some  pair  of  these  integers  whose  norms  have 
the  same  absolute  value  must  be  a  unit. 

This  method  of  determining  the  fundamental  unit  may  be 
shortened  by  observing  that,  if  c  +  d\/m  be  the  fundamental 
unit  of  &(\/m),  where  c  and  d  are  either  rational  integers  or 
rational  fractions  whose  numerators  are  odd  and  denominators 
2,  then  c  and  d  are  both  positive,  and  hence  no  equation  of  the 
form  3),  where  |a|<  2c,  can  have  as  a  root  a  unit  of  the  realm 
greater  than  1  and  less  than  c  +  d^Jm.  Therefore  the  funda- 
mental unit  is  a  root  of  the  first  equation  among  the  equations 


THE   UNITS    OF   THE    GENERAL   QUADRATIC   REALM.  423 

3),  arranged  in  ascending  values  of  \a\,  whose  roots  are  units 
of  k(\/m).  From  this,  we  see  that,  in  the  example  above,  it 
was  unnecessary  to  proceed  further  after  finding  i(i+V5)  as 

a  root  of  x2  —  x  —  i  =  o. 

The  number  of  equations  to  be  examined  may  also  be  reduced 
by  observing  that  we  must  have 

a2  +  4  53  o,  mod  m, 

if  an  equation,  whose  last  term  is  —  i,  is  to  have  as  a  root  a  unit 
of  k(y/m).  If  m  be  divisible  by  a  prime,  p,  of  the  form  411 —  I, 
this  relation  is  evidently  impossible,  for  it  requires  that  —  1  shall 
be  a  quadratic  residue  of  p.  Hence  the  fundamental  unit  of 
k(\/m)  can  not  have  — 1  as  norm,  if  m  be  divisible  by  a  prime 
of  the  form  4n —  /. 

§  5.     Pell's  Equation. 

It  will  be  at  once  recognized  that  the  determination  of  the  units 
of  a  real  quadratic  realm,  k(^/m),  is  equivalent  to  solving  Pell's 
Equation : 

x2  —  my2=±  1,  where  771  =  2  or  3,  mod  4, 
and  x2  —  my2  =  ±  4, 

or  x2  —  my2=±i,  where  m==l,  mod  4; 

furthermore  the  smallest  solution  will  give  the  fundamental  unit. 
The  general  problem  of  determining  an   integer   with   given 
norm,  H,  of  which  the  above  is  a  particular  case,  is  evidently 
equivalent  to  solving 

x2  —  my2  =  H. 

The  following  theorems  relating  to  Pell's  Equation  are  taken 
from  Chrystal's  Algebra,  Part  II,  p.  450,  and  the  reader  is  referred 
to  this  work  for  their  proof  and  for  the  complete  discussion  of 
this  subject.1  Confining  ourselves  now  to  solutions  in  which  x 
and  y  are  prime  to  each  other,  for,  if  x  and  y  have  a  common 
factor  r,  then  r2  must  be  a  factor  of  H  and  we  can  reduce  the 

1  See  also  H.  J.  S.  Smith :  pp.  192-200. 


424  THE    UNITS    OF   THE    GENERAL    QUADRATIC    REALM. 

equation  to  x'2 —  my'2  =  H',  where  H'=H/r2,  and  limiting  our 
discussion  to  the  case  |H|<|  V*»|,  we  have  the  following  theorem : 

Theorem  6.     The  equation 

x2  —  my2  =s  ±:  H 

where  m  andH  are  positive  integers  and  m  is  not  a  perfect  square, 
admits  of  an  infinite  number  of  solutions  provided  its  right-hand 
side  occurs  among  the  quantities  ( — i)nMn  belonging  to  the  devel- 
opment of  yjm  as  a  simple  continued  fraction,  zvhere  Mn  is  the 
(n-{-i)th  rational  divisor,  and  all  these  solutions  are  x  =  pn, 
y  =  qn,  where  pn/qn  is  the  11th  convergent  in  the  development 
of  V  w. 

Cor  i.     The  equation 

x2  — my2  =1  1 ) 

where  m  is  positive  and  not  a  perfect  square  always  admits  of  an 
infinite  number  of  integral  solutions,  all  of  which  are  furnished 
by  the  penultimate  convergents  in  the  successive  or  alternate 
periods  of  yjm. 

Cor  2.     The  equation 

x2  —  my2  =  — 1  2) 

where  m  is  positive  and  not  a  perfect  square  admits  of  an  infinite 
number  of  integral  solutions,  provided  there  be  an  odd  number  of 
quotients  in  the  period  of  "\/m,  and  all  these  solutions  are  fur- 
nished by  the  penultimate  convergents  in  the  alternate  periods 
of  \/m. 

If  there  be  an  even  number  of  quotients  in  the  period  of  V  m 
the  equation  has  no  integral  solution. 

If  p,  q  be  the  first  solution  of  1)  or  2)  and  we  have 

x~\-y  Vm=z  (P  ±  o.  Vm)ny 

where  n  takes  all  positive  values,  or  all  odd  positive  integral 
values.  Then  the  resulting  values  of  x,  y  are  all  solutions  of  1) 
or  2)  respectively. 


THE    UNITS   OF    THE    GENERAL   QUADRATIC   REALM. 


425 


For  the  discussion  of  the  equation 

x2  —  my2  ==±Hj 

where  H  is  greater  than  \/m,  the  reader  is  referred  to  Chrystal's 
Algebra,  Part  II,  p.  454. 

The  following  examples  will  illustrate  these  theorems : 

Ex.  1.     Determine  the  fundamental  unit  of  &(\/7)-     We  must  solve 
x2 —  7y2  = —  I,  if  possible,  and  if  not  possible,  then  x2 —  yy2=i. 
Expanding  \/y  in  a  continued  fraction  we  have 


1/7  =  2  + 


1+   1  + 


1        1 

~+  4  + 


+ 


ill 


which  gives  the  following  table,  where,  as  in  Chrystal,  n  is  number  of 
convergent,  an  the  wth  partial  quotient,  pn  the  numerator  of  the  nth 
convergent,  qn  the  denominator  of  the  nth  convergent,  Mn  the  (n+i)th 
rational  divisor. 


n 

* 

A 

In 

Mn 

1 

2 

2 

I 

3 

2 

1 

3 

I 

2 

3 

1 

5 

2 

3 

4 

1 

8 

3 

1 

5 

4 

37 

14 

3 

There  being  an  even  number,  4,  of  quotients  in  the  period  of  V7>  the 
equation  x2  —  7^2  =  —  1  has  no  solution  (Th.  6,  Cor.  2);  that  is,  the 
realm  k(yjy)  has  no  unit  with  negative  norm.  We  observe,  however, 
that  the  penultimate  convergent,  8/3,  in  the  period  of  V7  gives 

82  —  7  •  3"  =  1,  (Th.  6,  Cor.  1.) 

thatjs,  8  +  3V7,  8  —  3\/7,  —8  +  z\/J  and  —8  —  3V7"  are  units  of 
£(V7)>  8  +  3V7  being  the  fundamental  unit.  This  can  be  verified  by 
the  method  of  the  previous  section. 

Ex.  2.  Determine  the  fundamental  unit  of  ^(VI7)-  Expanding  yjiy 
in  a  continued  fraction,  we  have 

>/^  =  t+8i+   8T... 
which  gives  the  table,  there  being  only  one  quotient  in  the  period  of  VJ7- 


n 

an 

Pn 

In 

^„ 

I 

4 

4 

I 

1 

2 

8 

33 

8 

1 

Hence  the  equations 
x2- 


17  y 


4    and    x2 — I7y"  =  4 


426  THE   UNITS   OF   THE   GENERAL   QUADRATIC   REALM. 

have  no  solution,  but  the  equation 

x-  —  i/y'2  =  —  1 

has  the  solutions  x=  ±  4,  3;  =  ±  1,  and  4  +  V*7  1S  seen  to  be  the  funda- 
mental unit.  This  can  be  verified  by  seeing  that  among  the  roots  of  the 
equations 

x-  —  ax  ±  1  =  0 

where  \a\  <  2(4+  VT7)>  the  only  unit  of  k{\Jiy),  which  is  greater  than 
1,  is  4  +  Vi7.  _ 

Ex.  3.     Find  the  fundamental  units  of  the  realms  &("y/io),  £(-^11)  and 


CHAPTER   XIV. 
The  Ideal  Classes  of  a  Quadratic  Realm. 

§  i.    Equivalence  of  Ideals.1 

We  have  seen  (Chap.  XI,  Th.  5)  that  in  any  quadratic  realm, 
^(V«),  there  exists  for  every  ideal  a  an  ideal  m,  such  that  the 
product  am  is  a  principal  ideal. 

Attention  was  also  called  to  the  evident  fact  that  although  the 
particular  ideal  which  was  shown  to  have  the  desired  property 
was  the  conjugate  a'  of  a,  all  ideals  of  the  form  ct'(y),  where  (y) 
is  any  principal  ideal,  have  this  property. 

Since,  moreover,  if  a  and  16  be  any  two  ideals,  there  exists  in 
a  a  number  a  such  that  (a) /a  is  prime  to  b  (Chap.  XI,  Th.  11), 
it  is  evident  that  there  is  an  infinite  number  of  ideals  each  one 
prime  to  all  the  others  and  each  such  that  its  product  by  a  is  a 
principal  ideal ;  for,  if  ax  be  any  number  of  0,  then 

(a1)=aa1, 

where  at  is  an  ideal  having  the  desired  property.  By  the  above 
theorem  there  exists  in  a  a  number  a2  such  that 

(a2)~aa2, 

where  a2  is  prime  to  ax  and  is  evidently  an  ideal  having  the 
desired  property.  In  like  manner  there  exists  in  a  a  number  az 
such  that 

(a3)  =  aa3, 

where  a3  is  an  ideal  having  the  desired  property  and  prime  to 
0^2,  and  hence  to  each  of  them. 

Proceeding  in  this  manner,  it  is  evident  that  an  infinite  number 
of  ideals  exist  each  of  which  is  prime  to  all  of  the  others  and 
such  that,  when  multiplied  by  a,  the  product  is  a  principal  ideal. 

1Hilbert:  Bericht,  Cap.  VII. 

427 


428  THE   IDEAL   CLASSES   OF   A   QUADRATIC   REALM. 

We  see,  therefore,  that  the  ideal  m  need  not  contain  a'  as  a 
factor;  for  example, 

(2,  i+v=5)(3,  i+V-5)  =  (i+V-5)> 


(2,  i+y-5)(3,  i_v-5)  =  (i-V—  5), 

where  (3,  i  +  V — 5)  .an<3  (3,  1  —  V — 5)  are  prime  to 
(2,  1  —  V — 5)  and  to  each  other. 

From  the  fact  that  infinitely  many  ideals  give,  when  multiplied 
by  one  and  the  same  ideal,  products  which  are  principal  ideals, 
we  are  led  to  the  introduction  of  the  idea  of  the  equivalence  of 
ideals,  wThich  is  defined  as  follows : 

Tzvo  ideals,  a  and  fc,  are  said  to  be  equivalent  if  an  ideal  m 
exists  such  that  the  products  am  and  bm  are  both  principal  ideals. 

The  equivalence  of  a  and  B  is  expressed  symbolically  by  writing 

a'r^  6; 
that  they  are  not  equivalent  by  writing 

For  example,  as  we  have  seen  above,  the  product  of  each  of 
the  ideals  (3,  r~f-V—  5)  anc*  (3,  1—  V— -5)  by  the  ideal  (2, 
1  +V — 5)  is  a  principal  ideal;  hence  (3,  1  +V — 5)  *s  equiva- 
lent to  (3,  1 — V — 5) j  or  m  symbols 


(3,  1 +V— 5)^(3,  1— V— 5)- 
Likewise,  since  the  product  of  (2,  1 +V — 5)  by  itself  is  a 

principal  ideal,  (2,  1  +V — 5)  is  equivalent  to  each  of  the  two 

ideals  (3,  1  +V::r5)  and  (3,  1—  y/^). 

As  an  example  from  another  realm  k(  V —  17),  we  see  that 

(3,  1  +  V— '17)  ^  (ii,  4—  V— ~I7), 
for  it  can  be  easily  shown  that 

(11,  4-fy— T7)(n>  4— V— ^7)  =  (ii)> 


and       (11,  4+V—  I7)(3>  1  +V—  17)  =  U+V—  17)> 

If  a~b, 

then  by  the  definition  there  exists  an  ideal  c,  such  that 

QC=(ju), 


THE    IDEAL    CLASSES    CF   A   QUADRATIC   REALM.  429 

Multiplying  these  equations  respectively  by  b  and  a,  we  have 
abc=0)b=(»a; 
that  is,  if  a^h,  there  exist  two  integers,  /x  and  v,  such  that 

Furthermore,  if  a  and  b  be  any  two  ideals  and  there  exist  two 
integers,  /x  and  r,  such  that 

O)a=0*)b,  1) 

then  a  ^  b ; 

for  let  m  be  any  ideal  such  that  the  product  am  is  a  principal 
ideal  (y),  then  multiplying  i)  by  m,  we  have 

0)am=  (vy)  =  (/*)bm. 

But,  if  the  product  of  a  principal  ideal  and  another  ideal  be  a  prin- 
cipal ideal,  the  second  ideal  must  be  a  principal  ideal  also.  Hence 
bm  is  a  principal  ideal  and  consequently 

ct^b. 

We  may  therefore  define  the  equivalence  of  two  ideals  as  fol- 
lows, this  definition  being,  as  shown  above,  exactly  equivalent  to 
the  former  one: 

Two  ideals,  a  and  b,  are  equivalent  if  two  integers,  a  and  (3, 
exist  such  that 

a(/B)==»(a).> 

For  example,  we  have 

(i—V=~5)(3>  i+V=75)  =  (i+V::r5)(3)  i— V=5), 
whence  it  follows  that 

(3,  i  +V::::5)  ~  (3>  i  — V^)- 

We  shall  use  both  of  these  definitions  of  equivalence,  each 
having  some  advantages  of  its  own. 

Equivalences  between  ideals  obey  the  following  laws: 

i.  If  a^b  and  b  ^  c,  2) 

1  Hilbert :  Bericht,  p.  223. 


430  THE    IDEAL    CLASSES    OF   A   QUADRATIC   REALM. 

then  a  «— '  c, 

for  from  2)  it  follows  that  there  exist  integers  a,  /?,  y,  8,  such  that 

a(p)=Ha)  and  b(8)  =c(y), 
and  hence,  multiplying  these  equations  respectively  by  (8) and  (a), 

a(/?S)=c(ay), 
Therefore  a  /— '  C. 

ii.  If  a^b  and  c^b,  3) 

then  qc  r^  bb, 

for  from  3)  it  follows  that  there  exist  integers  a,  /?,  y,  8,  such  that 

a((3)=b(a)  and  c(8)=b(y), 
and  hence  ac(/?8)  =  bb(ay). 

Therefore  ac  ^  bb. 

If  a^h, 

then  from  ii  it  follows  immediately  that 

where  n  is  any  positive  rational  integer. 

The  original  definition  of  equivalence  given  above  is  that  used 
by  Dedekind,  the  second  is  equivalent  to  the  following,  which  is 
given  by  Hilbert  and  Weber: 

Every  number  of  a  realm,  *,  not  an  integer,  can  be  represented 
as  the  quotient  of  two  integers ;  that  is, 

If  now  we  look  upon  a  and  /?  as  principal  ideals  and  remove  all 
factors  common  to  (a)  and  (/?),  we  have 

(a)/G8)  =  a/6, 

a  representation  that  is  evidently  unique.     For  example,  let 


1  +  l/-  5 

/€=  

1-1/-5 


THE    IDEAL   CLASSES    OF   A   QUADRATIC   REALM.  43 1 

We  have 

(1  + 1/^5)  =  (2,  i  +  v^Xz,  1  +  v/:-j)  =  (3, 1  +  v^) 

(1  -  1/-  5)      (2,  1  +  V-  5)(3,  1  -  V-  5)      (3,  1  -  1/--5)  ' 

If  inversely  the  quotient  ct/b  of  two  ideals,  a  and  6,  where  a 
and  b  may  or  may  not  have  a  common  factor,  is  equal  to  the 
quotient  of  two  principal  ideals,  (a)  and  (/?)  ;  that  is,  if 

a/b=(a)/(/?), 
and  hence  may  be  taken  to  represent  in  the  above  sense  a  number, 
k  =  ol//3,  then  we  say  that  a  is  equivalent  to  b. 

For  some  purposes  it  is  useful  to  define  the  equivalence  of 
ideals  in  a  narrower  manner,  considering  a  equivalent  to  b  when 
and  only  when  a  number,  k,  whose  norm  is  positive  exists  such  that 

K  =  a/h; 

that  is,  when  two  integers,  a  and  /?,  whose  norms  have  the  same 
sign,  exist  such  that 

(/J)a=(a)6. 

This  definition  of  equivalence  will  evidently  be  essentially  dif- 
ferent from  the  original  one  when  and  only  when  the  realm  con- 
tains no  unit  whose  norm  is  negative.  In  quadratic  realms  this 
will  always  be  the  case  except  when  the  realm  is  real  and  the 
norm  of  the  fundamental  unit  is  —  I. 

In  general  this  definition  of  equivalence  is  identical  with  the 
original  one  in  all  realms  of  odd  degree. 

Examples.     Show  that  the  following  equivalences  hold 


I) 

(23,  8—  V—  5)^(7.  3+V— 5). 

2) 

(7.  i  +V  — 13)  ~  (2.  i+V— 13), 

3) 

(,i+f^y-(.), 

4) 

(2,  V— 10)  —  (5,  V— 10), 

5) 

(3,  1  — V— 14)2— (2,  V— 14), 

6) 

(5,  i+V26)-(2,V26), 

432  THE   IDEAL   CLASSES    OF   A   QUADRATIC    REALM. 

§  2.    Ideal  Classes. 

Since,  if  two  ideals,  a1}  a2,  be  equivalent  to  an  ideal  a,  they  are 
equivalent  to  each  other  (§  i,  i),  the  ideals  of  a  realm  can  be 
separated  into  classes  by  putting  two  ideals  into  the  same  or 
different  classes  according  as  they  are  or  are  not  equivalent  to 
each  other. 

The  system  of  ideals  composing  such  a  class  has  the  property 
that  every  ideal  in  it  is  equivalent  to  every  other  one  and  that  it 
consists  of  the  totality  of  all  ideals  which  are  equivalent  to  any 
one  of  the  ideals  composing  the  class. 

Such  a  class  is  called  an  ideal  class  and  is  denoted  by  a  Latin 
capital  letter. 

Any  ideal  a  of  a  class  A,  may  evidently  be  taken  as  the  repre- 
sentative of  the  class,  and  the  class  is  completely  determined  by  a. 

The  class  composed  of  all  principal  ideals  and  as  whose  repre- 
sentative we  can  take  (i),  is  called  the  principal  class  and  is 
denoted  by  i. 

If  alt  Q2  be  any  two  ideals  of  the  class  A,  and  blf  B2  be  any  two 
ideals  of  the  class  B,  then  since  from 

ctj.  ^  o2, 

and  &!  r^  fc2, 

it  follows  that 

aj)x^aj)2  (§i,ii), 

it  is  evident  that  all  ideals  of  the  form  ab,  where  a  and  b  are  any 
ideals  of  the  classes  A  and  B  respectively,  belong  to  a  single  class, 
C,  which  class  can,  however,  contain  infinitely  many  ideals  other 
than  the  products  ab. 

The  ideal  class  C  is  called  the  product  of  the  ideal  classes  A 
and  B  and  we  write 

C  =  AB. 

For  example,  we  have 

(3,  i  +  sF^S)  (2,  i  +  V^S)  =  (I  +  V~=~5), 
-whence  it  follows  that  the  product  of  the  classes  of  (3,  1  -f-  V  —  5)  and 
(2,     i  +  V  —  5)     is    the    principal    class.      But     (3,     1  +  V  —  5)     and 
<2,  1  +  V  —  5)  belong  to  the  same  class,  A.    Hence  we  have  A2=  1. 


THE   IDEAL   CLASSES    OF   A   QUADRATIC   REALM.  433 

The  product  of  any  ideal  class  A  by  the  principal  class  is  A  ; 
that  is, 

A-i=A. 

Inversely  from  AB  =  B 

it  follows  evidently  that  A  =  r. 

In  the  multiplication  of  ideal  classes  it  is  evident  from  the 
definition  of  the  product  of  two  classes  that  the  commutative  and 
associative  laws  hold;  that  is, 

AB  =  BA 

and  AB-C  =  A-BC. 

We  see,  therefore,  that  in  the  formation  of  the  product  of  any 
number  of  classes,  A19  A2,  •••,  Am,  the  order  in  which  the  classes 
are  taken  will  make  no  difference  in  the  final  result,  which  we 
denote  by  AXA2  •  •  •  Am. 

If  als  a2,  •••,  am  be  any  representatives  of  the  classes  Alt  A2, 
•  ••,  Am,  then  a^-dm  is  a  representative  of  the  class  AXA^ -'Am. 

If  each  of  the  m  factors  is  the  class,  A,  then  the  product  is 
called  the  wth  power  of  A  and  is  denoted  by  Am. 

We  have  A1  =  A 

and  A°  =  i. 

Theorem  i.     For  every  ideal  class  A  there  exists  one  and  only 

one  ideal  class  B  such  that  the  product  AB  is  the  principal  class.1 

Let  a  be  any  ideal  of  the  class  A  and  a  any  number  of  a.    Then 

ct6=(a),  1) 

where  b  is  an  ideal  whose  class  we  denote  by  B.     Then  from  i) 
it  follows  that 

AB  =  i.  2) 

If  now  a  class  C  other  than  B  exist  such  that 


AC=i,  3) 


1  Hilbert :  Bericht,  Satz  45. 
28 


434  THE   IDEAL   CLASSES   OF  A   QUADRATIC   REALM. 

we  have  from  2) 

ABC  =  C, 

and  hence,  making  use  of  3) 

B  =  C. 

The  theorem  is  therefore  proved. 

The  class  B  is  called  the  reciprocal  class  of  the  class  A  and  is 
denoted  by  A-1. 

It  is  evident  that  inversely  A  is  the  reciprocal  class  of  A'1. 

Defining  further  A-m  as  the  reciprocal  class  of  Am,  the  follow- 
ing laws  are  seen  to  hold  for  any  positive  integral  rational  expo- 
nents, r,  s. 

ArA8  =  Ar+s,  (Ar)s  =  Ars,  (AB)r  =  ArBr. 

Theorem  2.  //  A  be  any  ideal  class  and  b  any  ideal,  there 
exists  in  A  an  ideal  prime  to  b.1 

The  quotients  obtained  by  dividing  each  number,  a,  of  an  ideal 
a  by  a  are  evidently  ideals  that  belong  to  a  single  class. 

Among  them  can  be  found  an  ideal  prime  to  any  given  ideal 
b,  for  a  can  be  chosen  so  that  (a) /a  is  prime  to  b.  Hence  the 
theorem. 

§  3.    The  Class  Number  of  a  Quadratic  Realm. 

We  shall  now  show  that  the  number  of  ideal  classes  in  any 
given  quadratic2  realm  is  finite;  that  is,  there  exists  in  every 
quadratic  realm  a  system  of  ideals  finite  in  number  such  that  the 
product  of  any  ideal  of  the  realm  by  one  and  only  one  of  these 
ideals  is  a  principal  ideal.  Such  a  system  of  ideals  for  a  given 
realm  we  shall  call  a  complete  system  of  non-equivalent  ideals. 

The  number  of  ideals  composing  such  a  system,  that  is,  the 
number  of  ideal  classes  of  the  realm  is  denoted  by  h. 

To  prove  that  h  is  finite  we  need  the  following  theorem: 

Theorem  3.  In  every  ideal  a  there  exists  a  number  a  different 
from  0  and  such  that 

1  Dirichlet-Dedekind :  p.  579. 

2  This  theorem  holds  for  the  general  realm  of  the  wth  degree. 


THE    IDEAL   CLASSES   OF   A   QUADRATIC    REALM. 


435 


|*MUI»MV<*|, 

zvhere  d  is  the  discriminant  of  the  realm.1 

We  shall  distinguish  two  cases  according  as  the  realm  is  real 
or  imaginary. 

i.  Let  a  be  any  ideal  of  a  real  quadratic  realm,  k,  and 

a  basis  of  a,  where  o)1,  w2  is  a  basis  of  k.  Since  alt  a2  and  their 
conjugates,  a/,  a/  are  real  numbers,  k  being  a  real  realm, 
Ojjr  +  323^  a/'r  +  a2r3'  are  linear  forms  with  real  coefficients,  and 
their  determinant  can  easily  be  shown  to  be  different  from  o. 
Hence  by  Minkowski's  Theorem  (Chap.  XIII,  Th.  i)  there  exist 
rational  integers,  x0,  y0,  such  that 


K'.r0  +  a2'y0\  g  IV^a/  — a2a/| 

It  is  easily  seen  that  a,  =  a1x0  -\-a.2y0,  is  the  desired  number  of 
a,  for  if  a  —  axx0-\-a2y0,  then  a'  =  a1'x0-\- a2y0,  and  hence 
from  i) 

Moreover, 


that  is, 

«i     a2 

= 

ax     a2 

*>1        *>2 

and  hence 

\afi% 

t 

OA'I- 

=  / 

[a]V*|, 

|»[a]|s|»[o]V?|. 

ii.  The  realm  is  imaginary. 

Let  ax  =  pi  +  foj,    a2  =  p2  +  *<t2, 

where  pa,  p2,  o-j,  <r2  are  real  numbers  and  i  =  V —  i,  be  a  basis  of  a. 

Since  pi,  p2,  o^,  o\>  are  real  numbers,  whose  determinant  is  dif- 
ferent from  o,  there  exist  by  Minkowski's  Theorem  rational  in- 
tegers, x0,  y0,  such  that 


|Pl-r0  +  P23'o|  ^   I  VPl^2  /^l 


oV^o  +  <r2J0 1  g  j  Vpi°2  —  P2^i 


1  Hilbert :  Bericht,  Satz  46. 


436  THE   IDEAL    CLASSES    OF   A   QUADRATIC   REALM. 

We  shall  show  that 

a  =  axx0  +  a2y0 
is  the  desired  number. 
We  have 

a  =ct1x0  +a2y0  ^p^o  +  PsS^'fo^o +  »*?•)» 

a'  =  ck'x9  +  a2'y0  =  Plx0  +  p2y0  —  $  ( <rt*t  +  *2y0 ) , 

n[a]  =  (Plx0  +  p2y0)2  +  (atx9  +  <r2y0Y, 
and  hence 

MM  ^2|Plo-2— p2(Tx\. 

It  is  easily  seen,  moreover,  that 

I  «ia2'  —  a2a/ 1  =  2 1  Pl(r2  —  p^  | , 
whence  «[a]  ^  [flA' —  a2ax'|. 

We  have,  however,  as  in  i, 

\axa2'  —  a2a1'\  =  \n[a]Vd\, 
and  therefore 

n[a]  g  |»[q]  y~d\. 

Theorem  4.  There  exists  in  every  ideal  class  of  a  realm,  k, 
an  ideal  whose  norm  does  not  exceed  the  absolute  value  of  the 
discriminant  of  k} 

Let  A  be  any  ideal  class  and  \  an  ideal  of  the  reciprocal  class 
A-1.    By  the  last  theorem  there  exists  in  j  a  number,  1,  such  that 

|«H1*|»fflV3|.  2) 

But  (0=Kt,  3) 

where  a  is  an  ideal  belonging  to  the  class  reciprocal  to  A'1; 
that  is,  to  A. 

From  3)  it  follows  that 

\n[L]\=n[i]n[a], 
and  hence  from  2) 

»MslV3|- 

1  Hilbert :  Bericht,  Satz  50. 


THE    IDEAL    CLASSES   OF   A   QUADRATIC   REALM.  437 

Theorem  5.  The  number  of  ideal  classes  of  any  realm  is 
finite.1 

Since  every  ideal  is  a  divisor  of  its  norm,  we  shall  by  the  last 
theorem  obtain  at  least  one  representative  of  each  ideal  class  of 
any  given  realm,  k ;  that  is,  a  complete  system  of  non-equivalent 
ideals,  if  we  resolve  into  their  ideal  factors  all  positive  rational 
integers  which  are  less  than  |V^|>  where  d  is  the  discrimi- 
nant of  k. 

There  are  evidently  only  a  finite  number  of  rational  integers 
satisfying  this  condition  and  each  of  them  is  resolvable  into  only 
a  finite  number  of  ideal  factors.  The  number  of  ideals  of  k 
whose  norms  are  less  than  |V^|  *s  therefore  finite. 

Hence  the  number  of  ideal  classes  of  k  is  finite. 

The  last  two  theorems  enable  us  to  determine  the  number  of 
ideal  classes  of  any  quadratic  realm,  the  method  consisting  sim- 
ply in  determining  into  how  many  classes  the  finite  number  of 
ideals  fall,  whose  norms  are  less  than  |V^|.2 

We  shall  illustrate  this  method  of  determining  the  class  number 
by  several  examples.  This  we  do  the  more  readily  as  in  the 
solutions  of  these  examples  will  be  found  many  of  the  problems 
which  arise  in  reckoning  with  ideals. 

Our  task  then  being  to  ascertain  into  how  many  classes  the 
ideals  of  any  given  realm,  k,  fall,  whose  norms  are  ^|V^|,  it  is 
evident  that  this  will  be  accomplished,  if  we  determine  into  how 
many  classes  fall  the  prime  ideals  and  those  of  their  powers  and 
products  whose  norms  satisfy  the  given  condition. 

Having  determined  the  prime  ideals  whose  norms  are  g|V^| 
by  resolving  all  rational  primes  which  are  g|V^|  m*o  their  ideal 
factors,  we  next  determine  what  equivalences  exist  between  these 
ideals,  including,  of  course,  (1)  as  a  representative  of  the  prin- 
cipal class.  The  number  of  classes  given  by  these  prime  ideals 
and   (1)   having  been  determined,  it  remains  to  be  ascertained 

xHilbert:  Bericht,  Satz  50. 

2  This  method  of  determining  the  class  number  of  a  realm  is  applicable 
to  realms  of  higher  degree.  See  Hilbert :  Bericht,  p.  226;  also  "  Tafel 
der  Klassenanzahlen  fur  Kubische  Zahlkorper "  by  the  author. 


438  THE    IDEAL   CLASSES    OF   A   QUADRATIC   REALM. 

whether  any  powers  and  products  of  these  prime  ideals,  the  norms 
of  such  powers  and  products  being  ^| \/d\,  give  new  classes. 

The  solution  of  the  question  whether  or  no  two  given  ideals 
are  equivalent  will  be  discussed  in  full  in  connection  with  the 
numerical  examples. 

Theorem  6.     //  h  be  the  class  number  of  a  realm,  k,  the  hth 
power  of  every  ideal  class  is  the  principal  class.1 
Let  A  be  any  ideal  class  of  k. 
In  the  series 

A,  A2,  •  ••,  Ar,  •  ••, 

we  must  have  two  classes  the  same,  as 

Ar+e  =  Ar, 

and  hence  Ae=  I. 

If  Ae  be  the  lowest  power  of  A  which  gives  the  principal  class; 
then  the  classes 

A°=i,  A,  A*,  .-.,  A+*  4) 

are  all  different. 

If  B  be  a  class  different  from  all  the  classes  4),  then  the  classes 

B,  AB,  A2B,  -..,  A^B 

are  all  different  from  each  other  and  from  each  of  the  classes  4). 
Continuing  this  process,  we  see  that  h  is  a  multiple  of  e.  But  e 
was  the  exponent  of  the  lowest  power  of  any  class  that  gives  the 
principal  class. 

Hence  the  hth  power  of  every  class  of  k  is  the  principal  class. 

From  this  theorem  it  is  evident  that  the  hth  power  of  every 
ideal  is  a  principal  ideal. 

Ex.  1.     k(i).    Basis:  1,  i.    d  =  —  4. 

Each  class  must  contain  an  ideal  whose  norm  is  ^  |  \/ — 4  |,  that  is  ^2. 
We  shall  indicate  this  by  writing  n[ct]  i  |  V — 4l>  w[ct]  =1  or  2. 
We  have 

(2)  =  (l  +  02. 

The  only  ideals  whose  norms  satisfy  the  given  condition  are  therefore 
(1)   and  (i-f-*)»  both  of  which  are  principal  ideals.     There  is  therefore 

'Hilbert:  Bericht,  Satz  51. 


THE   IDEAL   CLASSES   OF   A   QUADRATIC   REALM.  439 

only  one  class,   the  principal  class.     Hence  h  =  1.     Therefore  the  ordi- 
nary unique  factorization  law  holds  in  k(i),  as  we  have  already  seen  to 
be  the  case. 
Ex.2.    k(\/~^z).    Basis:  1,  i(i  + V"1^)-    d  =  —  3 
n[a]^\  V^3|,  »[<*]  =  1. 

The  only  ideal  whose  norm  satisfies  the  given  condition  is  1,  hence  there 
is  only  one  class,  the  principal  class;  that  is, 


ftssrl. 

Ex.  3.    *(N/2). 

Basis:    1,  V2-    d  =  8 

re  have 

n[a]  ^  |  V8  j,  n[a]  =  1  or  2, 

(2)  =  V2)2. 

The  only  ideals  whose  norms  satisfy  the  given  condition  are   (1)    and 
(\/2),  both  of  which  are  principal  ideals. 
Hence 

ft  ±3  1-. 

Ex.  4.    &(V  —  5)-     Basis:    1,  V— 5-     d  —  —  20. 


w[a]  g  I  V  —20  |,  n[a]  =  i,  2,  3,  or  4. 
We  have 

(2)  =  (2,  i  +  V^5)2> 

(3)  =  (3,  i  +  V^"5)(3,    i-V^"5). 

We  have  now  to  determine  what  equivalences,  if  any,  exist  between  the 
ideals  (1),  (2,  I  +  V~ 5),  (3,  i  +  V^).  (3,  1  —  V^5)  and 
(2,  1  +  V  —  5)2>  these  being  all  the  ideals  whose  norms  satisfy  the  given 
condition.  We  see  at  once  that  (2,  1  +  V  —  5)2,  =  (2),  is  a  principal  ideal 
and  represents  therefore  with  (1),  the  principal  class. 

On  the  other  hand,  it  is  easily  shown  that  (2,  1  -J-  V  —  5)  is  a  non- 
principal  ideal,  for,  if  it  were  a  principal  ideal,  there  must  exist  an  integer, 
a,  =  x  +  yV  —  5,  such  that 

(a)  =  (2,  i  +  V=S). 

and  hence 

w[a]=w(2,  1  +  v^s); 

that  is,  two  rational  integers,  x,  y,  must  exist  such  that 

This  is,  however,  manifestly  impossible. 

Hence  (2,  1  +  V  —  5)  *s  a  non-principal  ideal  and  the  representative 
of  a  new  class,  which  we  shall  denote  by  A. 

We  have  already  proved  (§  1)  that  (3,  1  +  V  —  5)  ar,d  (3,  1  —  V  —  5) 
are  equivalent  to  (2,  1  +  V  —  5). 

They  belong  therefore  to  A,  and  all  ideals  of  &(V  —  5)  fall  into  two 
•classes,  1  and  A.     Hence  h  =  2.     It  will  be  observed  that  A*=l. 


440  THE    IDEAL    CLASSES    OF   A   QUADRATIC   REALM. 

Ex.  5-     *(V7>,    Basis:  1,  \f?.    d  =  28. 

«[<*]  ^  I  V28  I,      n[a]  =  1,  2,  3,  4,  or  5. 
We  have 

(2)  =  (2,1+  V7)2 

(3)  =  (3,  i  +  V7)(3,  i-V7) 

(5)  =  (5)-1 

The  ideals  to  be  considered  are  therefore  (1),  (2,  i-j-\/7)>  (3»  I  +  V7)» 
(3,  1- V7),  (5)  and  (2,  1  + V7)2;  of  these  (1),  (5)  and  (2,  1  +  V7)2 
belong  to  the  class  1. 

We  proceed  as  in  the  case  of  (2,  1  -f-  V  —  5)  in  the  last  example  to 
determine  whether  (2,  1  -\-  V7)  is  or  *s  not  a  principal  ideal.  In  order 
that  (2,  1  +  V7)  may  De  a  principal  ideal,  it  is  necessary  and  sufficient 
that  there  exist  an  integer  a,  =  x  -f-  y yjj,  such  that 

\n[a]  \=n(2,  1  +  V 7 )  ; 
that  is,  that  there  exist  rational  integers  x,  y,  such  that 

x2  —  7y2  =  2  or  —  2. 
We  see  that  x  =  3,  y  =  1  satisfy  this  condition.2     Hence 

(2,  i  +  V7)  =  (3  +  V7), 

a  principal  ideal,  3  +  V7  being  divisible  by  (2,  1  +  V7),  since  the  latter 
is  the  only  ideal  whose  norm  is  2.  We  can  in  like  manner  show  that 
(3>  1  +  V7)  is  a  principal  ideal,  for  x  =  2,  y  =  1  satisfy  the  condition 

*2  —  7y2  =  —  3 
whence 

(3,  1  +  V~7)  =  (2  +  V7)  or  r2  — V7). 
So  far  as  the  task  in  hand  is  concerned,  it  is  indifferent  to  which  of  the 
two  conjugate  principal  ideals,  (2  +  V7)  and  (2  —  V7)»  (3>  1  +  V7)  *s 
equal,  for  all  that  we  need  know  is  that  it  is  a  principal  ideal,  from 
which  it  follows  at  once  that  (3,  1  —  \/7)  1S  a  principal  ideal,  for  it 
belongs  to  the  class  reciprocal  to  that  of  (3,  1  +  V7)  since 

(3,  1  +  V7)  (3,  1  —  V7)  —  (1). 

It  is  easily  seen,  however,  that  2  +  V7  is  not  a  number  of  (3,  1  +  V7) 
while  2  —  \/7  does  enjoy  this  property.     Hence 

(3,    i  +  V7)  =  (2-V7), 
and 

(3,    i-V7)  =  (2  +  V7). 

All  the  ideals  of  k{\Jj)  whose  norms  are  i  |  V^  I  being  principal  ideals, 
we  have  h  =  1. 

1  This  denotes  that  (5)  is  a  prime  ideal. 

2  See  also  Chap  XIII,  §  5,  Pell's  Equation. 


THE    IDEAL    CLASSES   OF   A   QUADRATIC   REALM.  44 1 

We  are  assisted  in  determining  to  which  of  the  classes,  i,  A, 
A2,.-., A1,  if  any,  a  given  ideal  j  belongs  by  the  following 
theorem : 

Theorem  7.  //  a'  be  the  lowest  power  of  a  which  is  a  prin- 
cipal ideal,  a,  a2,  •  •  •,  a'  *->  1,  being  representatives  of  the  t  classes 

A,A\  ■•',A*=i,  5) 

and  }8  the  lowest  power  of  an  ideal  j  which  is  a  principal  ideal, 
then  in  order  that  j  may  belong  to  one  of  the  classes  5)  it  is  neces- 
sary that  t  shall  be  divisible  by  s,  and  furthermore,  if  this  condi- 
tion be  satisfied  and  t=t1s,  then  \  can  belong  to  none  of  the 
classes  5)  except  the  <f>(s)  classes  A*,  for  which  i=i1tly  and  ix  is 
prime  to  s. 

If  i-a% 

then  j*  ^  a**  ***  I  **  j8, 

whence  £  =  0,  mod  s; 

that  is,  t  divisible  by  s  is  a  necessary  condition  that  j  shall  belong 
to  one  of  the  classes  5). 

Furthermore,  if      *  j  r-*  a\ 


then 

Is  ~>  a8i  r^x^a*t 

whence 

si  sb  0,  mod  t,  sss  t^, 

and  therefore 

i  as  0,  mod  1 1 ; 

that  is, 

i  =  ixtx. 

Then 

j  ^  qMi, 

f^a2i^, 

jf , — ■  afiiti, 
\9  — '  affhti, 


442  THE   IDEAL    CLASSES    OF   A   QUADRATIC   REALM. 

from  which  it  follows,  since  no  two  of  the  ideals  j,  j2,  •  •  • ,  js  are 
equivalent,  that 

must  be  incongruent  each  to  each,  mod  t ;  that  is,  we  must  have 

where  /  and  g  are  any  two  of  the  integers,  i,  2,  •••,  s,  different 
from  each  other. 

Therefore  we  must  have      fix=^gix,  mod  s; 

that  is,  the  integers  ilt  2$lt  •••,  sit  must  form  a  complete  residue 
system,  mod  s,  which  can  be  the  case  only  when  i1  is  prime  to  ,?. 
Hence  in  case  j  should  belong  to  any  one  of  the  classes  5)  it 
is  possible  only  to  have 

i  ~  a***1, 

where  tx  =  t/s,  and  i1  is  prime  to  s. 

There  are  therefore  only  <f>(s)  of  the  classes  1)  to  which  it  is 
possible  for  j  to  belong. 

Ex.  6.  Let  a2*  be  the  lowest  power  of  a  which  is  a  principal  ideal, 
a,  a2,  •••,  cf*. — 'I,  representing  therefore  the  twenty-four  classes 

A,  A2,  -..,  A2i=i,  6) 

Let  f  be  the  lowest  power  of  j  which  is  a  principal  ideal. 

Since  24  is  divisible  by  6,  it  is  possible  for  j  to  belong  to  0(6)  =2,  of 
the  classes  6).  We  have  £  =  4,  and  those  of  the  classes  6)  to  which  it 
is  possible  for  j  to  belong  are  A*  and  A20. 

By  means  of  Th.  7  we  can  reduce  the  labor  of  determining  h ; 
for,  if  a  be  an  ideal  satisfying  Minkowski's  condition,  that  is, 
n[a]  ^|V^|j  and  Q*  the  lowest  power  of  a  that  is  a  principal 
ideal,  then 

a,  a2,  •••,  a*^  1, 

are  representatives  of  £  ideal  classes, 

^,  ^2,  ...,^*=i,  7) 

and,  as  we  have  seen  in  the  last  theorem,  h  is  a  multiple  of  t. 
Let  now  N  be  the  number  of  ideals  of  the  realm  that  satisfy 


THE   IDEAL   CLASSES   OF   A   QUADRATIC   REALM.  443 

Minkowski's  condition,  n  the  number  of  these  ideals  that  belong 
to  one  or  the  other  of  the  classes  7),  and  c  the  number  of  the 
known  classes  7)  that  have  found  representatives  among  the 
ideals  satisfying  Minkowski's  condition. 

The  t  classes  7)  must  evidently  have  representatives  among  the 
Ar  ideals  satisfying  Minkowski's  condition,  and  therefore,  since 
only  c  of  these  classes  have  yet  found  representatives  among  these 
ideals,  t  —  c  of  the  JV  —  n  of  these  ideals  whose  classes  have  not 
yet  been  determined  must  belong  respectively  to  the  t  —  c  classes 
whose  representatives  are  missing.  We  have  then  as  possible 
representatives  of  new  classes 

N  —  n — (t  —  c)  ideals,  and,  if 

N  —  n  —  (t  —  c)<t; 

that  is,  if  N  —  n-\-c<2t, 

it  follows,  since  h  must  be  divisible  by  t,  that 

h  =  t. 

In  particular,  if  N  <  2t, 

we  have  at  once  h  =  t. 

If  Ar  —  n  +  c<£2t, 

we  must  proceed  to  determine  whether  some  of  the  remaining 
ideals  belong  to  the  classes  7).  Let  j  be  one  which  is  found  to 
belong  to  none  of  the  classes  7)  and  let  \8  be  the  lowest  power  of  j 
which  is  a  principal  ideal. 

Then  j,  j2,  •••,  j*_1  are  representatives  of  the  s — 1  new  classes, 
B,  B2,  •  •  • ,  5S_1,  and  there  are  now  in  all  st  known  classes 

1,  A,  A\  ..-,  A**9 

B,  BA,  BA2,  •••,  BA*-\ 

8) 

B*-\  B*~K4,  B^A2,  ••-,  £s-M'-\ 

and  h  is  therefore  divisible  by  st. 


444  THE    IDEAL    CLASSES   OF   A    QUADRATIC    REALM. 

If  now  n  and  c  have  their  former  meaning  except  that  8)  are 
now  the  known  classes,  and  if 

N  —  n  +  c  <  2st, 

then  h  =  st. 

If,  however,  N —  n  +  c  <£  2^, 

we  proceed  as  before  to  determine  the  classes  to  which  the  remain- 
ing ideals  belong,  observing  always  whether 

N  —  n  +  c  <  2tf. 

If  we  find  one  that  belongs  to  none  of  the  classes  8),  we  proceed 
as  with  j. 


fl/-3i 


Ex.7.     KV—li).     Basis:   i,—^- ±,d=-3 


»[<*]  =  IV  — 31  I;   »[<*]  =  1,  2,   3,   4   or   5. 
We  have 


(3)  =  (3),  

(s)  =  (s.  Lbfc*)  (s.5=^> 


Since 

^  +  ^  +  8/ =4=2 


for  any  integral  values  of  x  and  y,  there  is  no  integer  of  &(V  —  31) 
whose  norm  is  2.    Hence 


(2,i+^.)~,( 


We  proceed  to  determine  the  lowest  power  of   (  2,  j 


that 


is  a  principal  ideal. 
We  have 


since  the  only  integer  of  &(V —  31).  whose  norm  is  4,  is  2,  and,  if 


then 


(,i±^ii)=(,), 


which  is  impossible. 


THE   IDEAL   CLASSES   OF   A   QUADRATIC   REALM.  445 

We   have 


since 

8  3=  l  +  V—  31 .  l  —  V  —  3\ 
2  2 

Hence  we  have  so  far  found  representatives,  1,    (  2,    5l J  ,    and 

^i+J^ZilV    (2y  1  +  ^~3IV^iof  three  classes  1,  A,  A2,  {A3=i). 

Therefore  h  is  divisible  by  3. 
Of  the  eight  ideals  satisfying  Minkowski's  condition,  (i),(  2,  — ■ — J, 


and   (  5> J     four   belong   to    these   classes    and    from 

we  see  that  (  2,  ^ J  belongs  to  A2,  and  hence  (  2,  £ 5-  J 

to  A. 

The  inequality  N  —  n  +  c  <  2t  is  now  seen  to  hold,  for  we  have  N  =  8, 
n  =  6,  c  =  3,  and  *  =  3,  and  it  is  evident  that  h  =  3.    The  classes  to  which 


(  S>  ^ )  and  (  5,  — ]  belong  are  easily  determined,  since 

and   3-f  V~Zl  is  a  number  of  both  (2,  1~V~3l\  and  (5,-^—). 


whence 


Therefore  (5>3  +  ^~31)  belongs  to  A,  and  ^IZUtlHj  to  A2. 

Ex.  8.    KV82).    Basis:  1,  V82.    ^  =  328. 

«[«]  =  V328  I ;  n[a]  =  1,  2,  3,  4,  5,  6,  7,  8,  9,  10,  II,  12,  13,  14,  15,  16,  17, 
or  18. 


446 


THE   IDEAL   CLASSES   OF  A   QUADRATIC   REALM. 


We  have 

(2)  =  (2,    V82)  (2,   V&) 

(3)  =  (3,  1  +  V82X3,  1-V82) 
(5)  =  (5) 

(7)  =  (7) 
(Il)  =  (ll,  4+ V82)(n,  4-V82) 
(13)  =(13,  2  +  V82)(i3,  2  — V&) 
(I7)  =  d7). 

We  must  now  determine  whether  (2,  V82)  is  a  principal  ideal.  To  do 
this  we  determine  whether  £(\/82)  contains  an  integer  whose  norm  is  2; 
that  is  whether  integral  values  of  x  and  y  can  be  found  satisfying  the 
equation 

x2  —  S2y2  =  2.  9) 

Using  Th.  6,  Chap  XIII,  and  developing  V82  as  a  continued  fraction, 
we  see  that 

1         1 


y/82  =  9-f 


18+18-^ 


and  have 


n 

an 

Pn 

fn 

Mn 

" 

9 

9 

I 

I 

2 

18 

163 

18 

I 

From  this  it  is  evident  that  9)  has  no  solution,  and  hence  that  (2,  V82) 
is  a  non-principal  ideal. 

From  this  development  of  V82,  it  is  also  evident  that  k(yJ82)  contains 
no  integers  with  norms  3,  5,  6,  or  7,  and  furthermore  9  +  V82  is  the 
fundamental  unit. 

That  &(V82)  contained  no  integers  with  norms  5  or  7  was,  of  course, 
already  shown  by  the  fact  (5)  and  (7)  are  principal  ideals.  We  have, 
however,  learned,  in  addition  to  the  fact  that  (2,  V82)  is  a  non-principal 
ideal,  that  (3,  1 -f- V82)  and  (3,  1  —  V82)  are  non-principal  ideals,  since 
&(V82)  contains  no  integer  with  norm  3,  and,  moreover,  that  neither  of 
the  products  of  these  last  two  ideals  by  (2,  V82)  can  be  a  principal  ideal, 
since  &(V82)  contains  no  integer  with  norm  6. 

We  shall  now  determine  into  how  many  classes  the  ideals,  which  have 
been  proved  to  be  non-principal,  fall. 

We  have  (2,  \/82)  as  a  representative  of  a  new  class,  A,  and  A2  =  1. 

Calculate  now  the  norms  of  a  few  integers  of  fc(V82).  We  have 
w[8+ V82]=  — 18. 

Hence  (18)  is  the  product  of  three  ideals  whose  norms  are  2,  3  and  3 
respectively.  Since  8+V82  is  a  number  of  (3,  1  —  V82)  and  not  of 
(3,  1  +  V82),  we  must  have 

(18)  =  (2f  V82)    (3,  1-V82)2. 


THE    IDEAL   CLASSES   OF   A   QUADRATIC   REALM.  447 

From  which  it  follows  that  (3,  I  —  V&O2  belongs  to  A,  and  (3,  1  —  V82) 
gives  a  new  class  B.     We  have  A  =  B2. 

But  n[i  —  V82]  =  —  81= — 3*,  and  1 — V&2  is  a  number  of  (3,  1 — V&2) 
and  not  of  (3,  1  +  V&2).    Hence 

(i-V82)  =  (3,  1-V82)*, 

and  we  see  that  we  now  have  four  classes  1,  B,  Br,  B3  (£*  =  1),  as  repre- 
sentatives of  which  among  the  ideals  satisfying  Minkowski's  condition, 
we  may  take  (1),  (3,  1  —  V82),  (2,  V&O  and  (3,  1 -f  V82).  We  have 
now  A'  =  28,  n  =  24,  c  =  4,  and  f  =  4,  and  hence  AT  —  n  -f  c  <£  2t ;  that  is, 
there  are  four  ideals,  the  factors  of  (11)  and  (13),  whose  classes  are  yet 
undetermined  and  we  have  found  representatives  of  all  of  our  four  known 
classes.  One  of  these  remaining  ideals  might  therefore  give  a  new  class 
and  we  should  have  h  =  8.  That  h  is  either  4  or  8,  we  now  know.  This 
is,  however,  easily  settled,  for  n{y  -J- V82I  —  —  ^  and  7  +  V82  is  a 
number  of  both  (3,  1  +  V82)  and  (11,  4  —  V82).     Hence 

(7  +  V82")  =  (3,  1  +  V82K11,  4-V&), 

and  (11,  4  —  V82)  belongs  to  the  class  B.    Therefore 

h  =  4- 

We  see  that  (11,  44-V82)  belongs  to  B3  and  from  the  fact  that  n[2+V82] 
=  —  78  =  —  2  •  3  •  13,  we  can  show  easily  that  (13,  2  -f  V82)  belongs  to 
B  and  (13,  2  — V82)  to  B3.  

Ex.  9.  Show  that  h=z  6  for  k  (V  —  26),  h  =  1  for  &(V  —  19),  /i  =  2 
for  ^(VI5)>  h  =  2  for  &(\/26),  A  =  4  for  &(V  —  34),  h  =  6  for 
fc(V^6i). 

The  labor  of  finding  h  by  this  method  can  be  reduced  by  using  another 
theorem,  due  also  to  Minkowski,  which  gives  a  smaller  limit  below  which 
the  norms  of  the  representatives  of  the  classes  must  fall,  thus  diminishing 
the  number  of  ideals  to  be  examined.  This  theorem  for  the  general  realm 
of  the  nth  degree  is  as  follows :  In  every  ideal  class  there  is  an  ideal,  a, 
such   that 


■M<(;?)'5!l»?l' 


where  n  is  the  degree  of  the  realm,  d  its  discriminant,  and  r  the  number 
of  pairs  of  imaginary  realms  which  occur  among  the  conjugate  realms, 

In  a  real  quadratic  realm,  we  have  n[a]  <  I  \  yjd '  \,  and  in  the  case 
of  fc(\/82)  need,  therefore,  to  examine  only  those  ideals  whose  norms  are 
less  than  10. 

It  will  be  noticed  that  we  did  find,  as  representatives  of  all  classes,  ideals 
whose  norms  satisfied  this  condition. 

1  Minkowski :  Diophantische  Approximationen,  p.  185.  See  also  "  Tafel 
der  Klassenanzahlen  fur  Kubische  Zahlkorper "  by  the  author  for  its 
application  to  cubic  realms. 


448  THE   IDEAL   CLASSES   OF   A   QUADRATIC   REALM. 

For  a  table  giving  the  class  numbers  of  quadratic  realms,  their  funda- 
mental units  and  other  data,  see  J.  Sommer:  Vorlesungen  uber  Zahlen- 
theorie. 

This  table  extends,  for  imaginary  realms,  to  ra  = —  97,  and,  for  real 
realms,  to  m  =  101.  This  book  should  be  consulted  by  those  who  wish 
to  pursue  the  subject  further. 

The  class  number  of  a  realm  can  also  be  expressed  by  means  of  an 
infinite  series.  See  Hilbert :  Bericht,  Cap.  VII  and  §79;  also  Dirichlet- 
Dedekind:  §184. 

We  shall  close  this  chapter  with  a  theorem  that  gives  important 

information  regarding  the  class  number  of  a  realm  in  a  certain 

special  case.     For  its  proof,  we  shall  need  two  theorems,  the 

second  of  which  throws  additional  light  upon  the  question  whether 

the  norm  of  the  fundamental  unit  of  a  real  quadratic  realm  is 

1  or  — 1. 

Theorem  8.  Every  number,  a,  of  a  quadratic  realm,  &(\/m), 
whose  norm  is  1,  can  be  represented  as  the  quotient,  y/y',  of  two 
conjugate  integers,  y,  y  ,  of  the  realm} 

We  have  seen  that  a  can  be  put  in  the  form 

a  +  bat 


c 

where  1,  w  is  a  basis  of  the  realm  and  a,  b  and  c  are  rational 
integers.  Let  y  —  x-\-  ym,  where  x  and  y  are  rational  integers  to 
be  determined,  and  let  the  rational  equation  of  which  w  is  a  root  be 

x2  +  ex  +  f  =  o. 
Put 

a  -f-  boy        x  4-  \'o)  N 

! —g  — L^_  .  10) 

C  X  +  To/ 

Making  use  of  the  relations  »-)-«'==  —  e,  and  ww'  =  /,  we 
have  from  10),  as  the  equations  that  x  and  3!  must  satisfy, 

(o  —  c)x-\-  (bf  —  ae)x  =  o, 

") 

bx —  (a  +  c)y=0. 

These  equations  evidently  have  a  solution  different  from  x  =  (\ 
1  See  Hilbert :  Bericht,  Satz  90. 


THE    IDEAL   CLASSES   OF   A   QUADRATIC   REALM.  449 

y  =  o  when  and  only  when  the  determinant,  D,  of  their  coefficients 
is  o,  and,  if  D  =  o,  they  have  an  infinite  number  of  solutions 
jfss=:r.Xii  y  =  ryi,  where  xx,  yt  is  any  particular  solution  different 
from  o,  o,  and  hence  have  an  infinite  number  of  integral  solutions, 
for  we  can  choose  r  so  that  rxlf  ry±  are  integers. 
We  have 

D  =  —  a2  +  abe  —  b2f  +  c2  =  —  n [a]  •  c2  +  c2  =  o, 

since  n[a]  =  I.     Hence  the  equations  n)  have  an  infinite  number 
of  integral  solutions  and  the  theorem  is  therefore  proved. 
-    As  a  particular  solution  of  n),  we  may  take  x  =  a-\-c,  y  =  b, 
all  integral  solutions  then  being  of  the  form 

s(a  +  0  sb 

*-._-■  ,   y=  -j, 

where  ^  and  t  are  rational  integers  and  t  a  common  divisor  of 
a-\-  c  and  b. 

We  can,  of  course,  take  a,  b  and  c  without  a  common  divisor, 
and  then  have  also  a  prime  to  b,  since  n[a]  =  i. 

Ex.    Let  a  = — .     We  have  a  =  2,  b=i,  c  =  3,  and  hence 

3 

3  5—  V"— S 

Theorem  9.  //  f/i^  discriminant,  d,  of  a  real  quadratic  realm, 
k(y/m),  be  divisible  by  a  single  prime  number,  the  norm  of  the 
fundamental -unit  of  the  realm  is  —  I.1 

In  order  that  d  may  be  divisible  by  a  single  prime  number,  we 
must  have  m  =  2,  or  a  prime  =  1,  mod  4. 

Let  c  be  the  fundamental  unit  of  &(  Vw)- 

If  w[e]=i,  by  Th.  8  there  would  exist  an  integer,  y,  of 
k(-\/m)  such  that 

e  =  ?-.  12) 

r 

Then  from  12)  it  would  follow  that 

(y)  =  (/), 

1  Hilbert :   Bericht,  p.  294. 
29 


450  THE   IDEAL   CLASSES    OF   A    QUADRATIC    REALM. 

and  hence  that  (y)  is  either  an  ambiguous  ideal  (p.  347),  an 
ambiguous  ideal  multiplied  by  a  rational  principal  ideal  (a),  or 
(a).  Since,  however,  d  is  divisible  by  the  single  prime  m,  the 
realm  contains  only  one  ambiguous  prime  ideal  (\/w),  which  is 
therefore  the  only  ambiguous  ideal  of  the  realm.  Hence,  we 
must  have 

(y)='(Vw),  (aV»)  or  (a), 

and  therefore  y^^ym,  rja^/tn  or  rja, 

where  77  is  a  unit.     But  we  have  then  from  12) 

or 


and  hence  c  =  —  rj2  or  rf, 

from  which  it  would  follow  that  e  is  not  the  fundamental  unit,  as 
was  assumed.  Hence  the  assumption  that  n  [e]  =  1  is  untenable, 
and  the  theorem  is  proved. 

The  realms  k{^/2),  k{-\J$)  and  &(\/i7),  whose  fundamental  units 
have  been  found  to  be  1  +  y 2,  £(i  +  \/5)  and  4+^17  respectively, 
will  illustrate  the  truth  of  this  theorem. 

Theorem  10.  //  the  discriminant  of  a  quadratic  realm,  k  (  ^/m), 
be  divisible  by  a  single  prime  number,  the  class  number,  h,  of  the 
realm  is  odd.1 

Assume  h  to  be  even.  Then  there  is  in  the  realm  certainly  one 
non-principal  ideal,  j,  whose  square  is  a  principal  .ideal ;  that  is, 
j2  r^  1.  But  we  have  also  jj'  —  1,  and  hence  j  ^  f ;  that  is,  there 
exist  integers,  a,  /?,  of  the  realm  such  that 

(a)i=(/J)f.  13) 

From  13)  we  have  n[(a)]=n[(p)]9  whence  a//3,  =  /c,  is  a 
number  of  the  realm  whose  norm  is  ±  1.  When  k(-\/m)  is 
imaginary,  we  have  *[*]=* I,  and  when  k(\/m)  is  real  and 
n[e]  = — ■  1,  where  e  is  the  fundamental  unit,  we  have  either 
n[K.]=ii,  or  n[ac]  =  i.  By  Th.  8  we  can  put  K  =  y/y,  or 
CK  =  y/y',  according  as  h[k]  =  1  or.  —  1,  y  and  y'  being  conju- 
gate integers  of  the  realm.     In  both  cases,  we  have 

'Hilbert:  Bericht,  Hiilfsatz  13. 


THE   IDEAL   CLASSES   OF   A   QUADRATIC   REALM.  45  I 

and  hence  from  13)  (y)j=  (y')i'»  as  a  consequence  of  j2  <-*>  1, 
where  j  is  a  non-principal  ideal ;  that  is,  as  a  consequence  of  h  even. 
Hence  (y)j  is  either  an  ambiguous  ideal,  an  ambiguous  ideal 
multiplied  by  a  rational  principal  ideal  (a),  or  (a).  Since,  how- 
ever, when  m  =  2,  or  a  prime  =  1,  mod  4,  the  realm  contains  no 
ambiguous  ideal  other  than  (Vm)  (see  proof  of  Th.  9),  and,  in 
in  the  case  of  k(i),  the  only  ambiguous  ideal  is  (1  +  *).  We  see 
that  in  all  cases  (y)j  must  be  a  principal  ideal,  and  hence  j  a 
principal  ideal.  But  this  renders  untenable  our  assumption  that 
h  is  even.     Hence  h  is  odd. 


The  realms  k(i),  k(^/ — 3),  k(yf2)  and  k{^J — 31),  whose  class 
numbers  were  found  to  be  1,  1,  1  and  3  respectively,  will  illustrate  the 
truth  of  this  theorem. 

It  is  evident  that  in  determining  the  class  number  of  a  realm, 
satisfying  the  conditions  of  Th.  10,  we  can  use,  since  h  must  be 
odd,  instead  of  the  inequality  N  —  n  -j-  c  <  2t,  the  inequality 
N  —  n  -f-  c  <  3/,  thus  shortening  the  work  still  further.  Making 
use  of  this  in  Ex.  7,  it  is  unnecessary  to  determine  the  class  to 


which  belongs  (  2, j. 


INDEX. 


Numerals  refe 

Ambiguous  ideal,  347. 

Appertains,  exponent  to  which  an  in- 
teger, 99,  393. 

Associated  integers,  in  R,  9  ;  in  k(i), 
163;  in  k(V — 3),  223;  in  kiV^), 
246. 

Basis,  of  k(i),  159-161  ;  of  £  V — 3), 
220;  of  k(V2),  232;  of  k(V  —  5), 
245  ;  of  k(Vtn),  284-287,  determi- 
nation, 289-292 ;  of  ideal,  293-295, 
determination,    351-355. 

Biquadratic  residues  and  reciprocity 
law,  205-217. 

Character  of  an  integer,  quadratic,  in 
R,  121,  in  k(i),  212;  biquadratic, 
209,  212 

Classes,  ideal,  definition,  432 ;  prin- 
cipal class,  432 ;  product  of,  432 ; 
reciprocal,  434. 

Classification  of  the  numbers  of  an 
ideal  with  respect  to  another  ideal, 
326-330. 

Class  number  of  a  realm,  definition, 
434;  is  finite,  437;  determination, 
437-448,  45 1- 

Congruences,  definition,  31,  297,  323  ; 
elementary  theorems,  32-37 ;  323- 
326 ;  of  two  polynomials,  57,  370 ; 
of  condition,  59-61,  369-372 ;  of 
first  degree  in  one  unknown,  68-70, 
375-38o ;  equivalent,  62-64,  372, 
373 ;  transformations,  62-64,  372, 
374 ;  equivalent  systems,  64 ;  of  nth 
degree  in  one  unknown,  preliminary 
discussion,  66-68,  374,  375,  root,  66, 
374,  with  prime  modulus,  88-90, 
385-387,  composite  modulus,  95-97, 
39i,  392;  multiple  roots,  definition, 

452 


r  to  pages. 

89,  386,  determination,  93,  94,  386  ; 
limit  to  number  of  roots,  89,  386  ; 
x4><-m> — i=o,  mod  m,  90  ;  x<t>(m) — 1 
==  o,  mod  m,  387,  388 ;  common 
roots,  92,  93,  389;  binomial,  110- 
112,  primitive  and  imprimitive 
roots,  in;  xn^=b,  mod  p,  114- 
116,  Euler's  criterion,  115;  of  sec- 
ond degree  with  one  unknown,  119- 
121  ;  solution  of  x2^^. —  1,  mod  p, 
by  means  of  Wilson's  theorem,  129, 
130;  in  k(i),  180,  of  condition,  190. 
Conjugate,  numbers,  4 ;  realm,  4. 

Dirichlet's  theorem  regarding  infinity 
of  primes  in  an  arithmetical  pro- 
gression, 11. 

Discriminant,  of  k(i),  161  ;  of 
k(V —  3),  221;  of  k(y/2),  232;  of 
k(V^S),  245;  of  k(Vm),  287, 
288 ;  of  number,  284. 

Divisor,  greatest  common,  in  R,  16, 
18,  25;  in  k(i),  173;  of  two  ideals, 
310-313,  318;  discussion  of  defini- 
tion, 252. 

Divisors,  of  integers  in  R,  number  of, 
23,  sum  of,  24 ;  of  ideal,  number  of, 
318. 

Equivalence    of    ideals,    427-431  ;    in 

narrower  sense,  431. 
Eratosthenes,  sieve  of,   10. 
Euler's    criterion    for    solvability    of 

xn^=b,  mod  p,  115,  122. 

Factorization  of  a  rational  prime  de- 
termined by  (d/p),  in  k(i),  179;  in 
k(V —  3),  229;  in  fe(Vm),  347, 
348. 

Fermat's  theorem,  57  ;  as  generalized 


INDEX. 


453 


by    Euler,    57  ;    analogue    for    k(i), 

189  ;  analogue  for  ideals,   368,   369. 

Frequency  of  the  rational  primes,   II, 

Galois  realm,  281. 
Gauss'  lemma,    130. 
Generation   of  realm,   3. 

Ideal  numbers,  necessity  for,  253  ; 
nature  explained,  254-257 ;  Kum- 
mer's,  267. 

Ideals,  definition,  257,  293 ;  numbers 
of,  293 ;  basis  of,  293-295  ;  can- 
onical basis  of,  294 ;  determination 
of  basis,  298-301  ;  numbers  defin- 
ing, 295  ;  symbol  of,  257,  295  ;  in- 
troduction of  numbers  into  and 
omission  from  symbol,  258,  295, 
296 ;    principal    and    non-principal, 

260,  261,  297 ;  conjugate,  301  ; 
basis  of  conjugate,  301  ;  equality  of, 
258,    259,    302 ;    multiplication    of, 

261,  262,  302,  303;  divisibility  of, 
263,  303  ;  common  divisor  of,  303  ; 
prime,  263-265,  304 ;  norm  of,  326- 
338,  351. 

Imprimitive  numbers,  see  primitive 
numbers. 

Incongruent  numbers,  complete  sys- 
tem of,  in  R,  34;  in  fe(i'),  182-185; 
in  k(Vm),   326. 

Index,  of  a  product,  106,  399;  of  a 
power,   106,   399. 

Indices,  definition,  105,  399;  system 
of,  106,  399 ;  solution  of  congru- 
ences by  means  of,  1 08-1 10,  400- 
402. 

Integers,  of  R,  7,  23;  absolute  value 
in  R,  7,  33;  of  k(i),  157;  of 
k(V  —3),  219;  of  k(V2),  231;  of 
k(V  —  5),  245;  of  k(Vm),  284- 
287;  general   algebraic,    1,   275-279. 

Legendre's  symbol,  127. 

Multiple,  least  common,  in  R,  25  ;  of 
two  ideals,  310-312,  318. 


Non-equivalent  ideals,  complete  sys- 
tem  of,   434. 

Norm,  of  a  number,  in  fe(i'),  156;  in 
k(V^3),  218,  221  5  in  k( V2),  231  ; 
in  k(V  —  5),  245;  in  k(Vrn),  283; 
of  an  ideal,  definition,  326,  337, 
value,  330,  determination,  351 ;  of  a 
product  of  ideals,  334 ;  of  a  prin- 
cipal ideal,  337  ;  of  a  prime  ideal, 
338. 

Numbers,  algebraic,  definition,  1  ;  de- 
gree of,  1  ;  conjugate,  4 ;  rational 
equation  of  lowest  degree  satisfied 
by,  2,  273;  of  R,  7;  of  k(i),  155; 
of  k(V~^3),  218;  of  fc(V2),  231; 
of  k(V  —  5),  245;  of  the  general 
realm,   271-279;  of  fe(Vw),  281. 

Number  class,  rational  modulus,  32, 
33  ;  ideal  modulus,  324. 

Pell's  equation,  423-426. 

0- function,  in  R,  definition,  37,  gen- 
eral expression,  38,  44,  53,  product 
theorem,  45,  summation  theorem, 
46,  75,  of  higher  order,  54 ;  in 
k(i),  185-188;  for  ideals,  definition, 
358,  expression  for  power  of  prime 
ideal,  359,  general  expression,  359-- 
362,  366,  367,  summation  theorem, 
362,  363,  367,  product  theorem,  360, 
361,  of  higher  order,  367. 

Polynomials  in  a  single  variable,  268- 
271. 

Polynomials  with  respect  to  a  prime 
modulus,  reduced,  62 ;  degree  of, 
76 ;  divisibility  of,  76,  380 ;  com- 
mon divisor  of,  76,  380 ;  common 
multiple  of,  76,  380 ;  unit,  77, 
381  ;  associated,  77,  381  ;  primary, 
78,  381  ;  prime,  78,  381  ;  determina- 
tion of  prime,  78,  381,  382;  congru- 
ence with  respect  to  a  double 
modulus,  81  ;  unique  factorization 
theorem  for,  82-87,  382-385  ;  divi- 
sion  of  one  by   another,    382. 

Power  of  a  prime  by  which  m !  is 
divisible,  26. 


454 


INDEX. 


Primary   integers   of   k(i),    193-196. 

Prime  factors,  resolution  of  an  ideal 
into,  348-350. 

Prime  ideals,  of  k(V  —  5),  263-265; 
of  k(Vm),  definition,  304,  deter- 
mination and  classification,  339- 
348. 

Prime  numbers,  of  R,  definition,  9, 
infinite  in  number,  10  ;  of  k(i),  defi- 
nition, 165,  classification,  177;  of 
k(V  —  3),  definition,  223,  classifi- 
cation, 227-230;  of  k(V2),  defi- 
nition, 235,  classification,  238-240  ; 
of  k(V  —  5),  246,  247. 

Primitive  numbers,  of  k(i),  157  ;  of 
k(V~^3),  218;  of  the  general 
realm,  274,  275;  of  k(V*n),  282, 
283  ;  with  respect  to  a  prime  ideal 
modulus,   398. 

Primitive  root,  definition,  100;  deter- 
mination, 112;  of  prime  of  form 
224-i,  151;  of  prime  of  form 
49+i    is   2,    152. 

Realm,  definition,  3  ;  generation,  3  ; 
degree,  4 ;  conjugate,  4 ;  number 
defining,  4,  280 ;  number  generating, 
4. 

Reciprocity  law,  for  quadratic  resi- 
dues, in  R,  135;  in  k(i),  201-205; 
determination  of  value  of  (a/p)  by 
means  of,  144;  other  applications 
of,  149;  for  biquadratic  residues, 
210,  215-217. 

Residue,  odd  prime  moduli  of  which 
an  integer  is  a  quadratic,  128,  145, 
147  ;  prime  moduli  of  which  —  1,  is 
a  quadratic,  128;  prime  moduli  of 
which   2  is   a  quadratic,   133. 

Residue  system,  complete,  in  R,  33, 
34;    in    k(i),    182-185;    in    fe(Vm), 


326;  reduced,  in  R,  37;  in  k(i), 
185,  in  fe(Vw),  358. 

Residues  of  powers,  definition,  98, 
392 ;  complete  system  of,  98,  393 ; 
law   of  periodicity,    100. 

Residues,  n-ic,  116;  quadratic,  in  R, 
131,  in  k(i),  196-201  ;  quadratic 
non-,  121  ;  determination  of  quad- 
ratic, 124  ;  with  respect  to  a  series 
of  moduli,  integer  having  certain, 
70 ;  cubic,  250 ;  biquadratic,  205- 
217. 

Sub-realm,    157. 

Symbol,  Legendre's,  127 ;  for  ideal, 
257,  295. 

Unit  ideal,  of  k(V  —  5),  263;  of 
k(Vm),  304. 

Unit,  fundamental,  of  fe(V2),  233; 
of  k(Vm),  definition,  420;  deter- 
mination,  420-426. 

Units,  of  R,  8;  of  k(i),  163;  of 
fc(V  —  3),  222;  of  fc(V2),  232- 
23S;  of  k(V  —  5),  246;  of  fe(Vm), 
definition,  403,  realm  imaginary, 
404,   realm  real,   405-426. 

Unique  factorization  theorem,  in  R, 
12;  in  k(i),  167,  174,  graphical 
discussion  of,  169;  in  k(V2),  236, 
237;  in  k(V—3),  226;  in  k(V — 5), 
failure  of,  247-253,  necessity  for, 
253,  restoration  in  terms  of  ideal 
factors,  265,  266  ;  realms  in  which 
original  method  of  proof  holds, 
248-250;  for  ideals  in  fe(Vw), 
305-317. 

Wilson's  theorem,  91  ;  as  generalized 
by  Gauss,  91  ;  analogue  for  ideals, 
388,  389- 


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